title: Space (mathematics)

text: Two equivalence relations between geometric figures were used: congruence and similarity. Translations, rotations and reflections transform a figure into congruent figures; homotheties — into similar figures. For example, all circles are mutually similar, but ellipses are not similar to circles. A third equivalence relation, introduced by Gaspard Monge in 1795, occurs in projective geometry: not only ellipses, but also parabolas and hyperbolas, turn into circles under appropriate projective transformations; they all are projectively equivalent figures.


title: Space (mathematics)

text: The relation between the two geometries, Euclidean and projective, shows that mathematical objects are not given to us "with their structure". Rather, each mathematical theory describes its objects by "some" of their properties, precisely those that are put as axioms at the foundations of the theory.


title: Space (mathematics)

text: Distances and angles cannot appear in theorems of projective geometry, since these notions are neither mentioned in the axioms of projective geometry nor defined from the notions mentioned there. The question "what is the sum of the three angles of a triangle" is meaningful in Euclidean geometry but meaningless in projective geometry.


title: Space (mathematics)

text: A different situation appeared in the 19th century: in some geometries the sum of the three angles of a triangle is well-defined but different from the classical value (180 degrees). Non-Euclidean hyperbolic geometry, introduced by Nikolai Lobachevsky in 1829 and János Bolyai in 1832 (and Carl Friedrich Gauss in 1816, unpublished) stated that the sum depends on the triangle and is always less than 180 degrees. Eugenio Beltrami in 1868 and Felix Klein in 1871 obtained Euclidean "models" of the non-Euclidean hyperbolic geometry, and thereby completely justified this theory as a logical possibility.


title: Space (mathematics)

text: This discovery forced the abandonment of the pretensions to the absolute truth of Euclidean geometry. It showed that axioms are not "obvious", nor "implications of definitions". Rather, they are hypotheses. To what extent do they correspond to an experimental reality? This important physical problem no longer has anything to do with mathematics. Even if a "geometry" does not correspond to an experimental reality, its theorems remain no less "mathematical truths".


title: Space (mathematics)

text: A Euclidean model of a non-Euclidean geometry is a choice of some objects existing in Euclidean space and some relations between these objects that satisfy all axioms (and therefore, all theorems) of the non-Euclidean geometry. These Euclidean objects and relations "play" the non-Euclidean geometry like contemporary actors playing an ancient performance. Actors can imitate a situation that never occurred in reality. Relations between the actors on the stage imitate relations between the characters in the play. Likewise, the chosen relations between the chosen objects of the Euclidean model imitate the non-Euclidean relations. It shows that relations between objects are essential in mathematics, while the nature of the objects is not.


title: Space (mathematics)

text: The word "geometry" (from Ancient Greek: geo- "earth", -metron "measurement") initially meant a practical way of processing lengths, regions and volumes in the space in which we live, but was then extended widely (as well as the notion of space in question here).


title: Space (mathematics)

text: According to Bourbaki, the period between 1795 ("Géométrie descriptive" of Monge) and 1872 (the "Erlangen programme" of Klein) can be called the golden age of geometry. The original space investigated by Euclid is now called three-dimensional Euclidean space. Its axiomatization, started by Euclid 23 centuries ago, was reformed with Hilbert's axioms, Tarski's axioms and Birkhoff's axioms. These axiom systems describe the space via primitive notions (such as "point", "between", "congruent") constrained by a number of axioms.


title: Space (mathematics)

text: Analytic geometry made great progress and succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups. Since that time, new theorems of classical geometry have been of more interest to amateurs than to professional mathematicians. However, the heritage of classical geometry was not lost. According to Bourbaki, "passed over in its role as an autonomous and living science, classical geometry is thus transfigured into a universal language of contemporary mathematics".


title: Space (mathematics)

text: Simultaneously, numbers began to displace geometry as the foundation of mathematics. For instance, in Richard Dedekind's 1872 essay "Stetigkeit und irrationale Zahlen" ("Continuity and irrational numbers"), he asserts that points on a line ought to have the properties of Dedekind cuts, and that therefore a line was the same thing as the set of real numbers. Dedekind is careful to note that this is an assumption that is incapable of being proven. In modern treatments, Dedekind's assertion is often taken to be the definition of a line, thereby reducing geometry to arithmetic. Three-dimensional Euclidean space is defined to be an affine space whose associated vector space of differences of its elements is equipped with an inner product. A definition "from scratch", as in Euclid, is now not often used, since it does not reveal the relation of this space to other spaces. Also, a three-dimensional projective space is now defined as the space of all one-dimensional subspaces (that is, straight lines through the origin) of a four-dimensional vector space. This shift in foundations requires a new set of axioms, and if these axioms are adopted, the classical axioms of geometry become theorems.


title: Space (mathematics)

text: A space now consists of selected mathematical objects (for instance, functions on another space, or subspaces of another space, or just elements of a set) treated as points, and selected relationships between these points. Therefore, spaces are just mathematical structures of convenience. One may expect that the structures called "spaces" are perceived more geometrically than other mathematical objects, but this is not always true.


title: Space (mathematics)

text: According to the famous inaugural lecture given by Bernhard Riemann in 1854, every mathematical object parametrized by "n" real numbers may be treated as a point of the "n"-dimensional space of all such objects. Contemporary mathematicians follow this idea routinely and find it extremely suggestive to use the terminology of classical geometry nearly everywhere.


title: Space (mathematics)

text: Functions are important mathematical objects. Usually they form infinite-dimensional function spaces, as noted already by Riemann and elaborated in the 20th century by functional analysis.


title: Space (mathematics)

text: While each type of space has its own definition, the general idea of "space" evades formalization. Some structures are called spaces, other are not, without a formal criterion. Moreover, there is no consensus on the general idea of "structure".


title: Space (mathematics)

text: According to Pudlák, "Mathematics [...] cannot be explained completely by a single concept such as the mathematical structure. Nevertheless, Bourbaki's structuralist approach is the best that we have."


title: Space (mathematics)

text: We will return to Bourbaki's structuralist approach in the last section "Spaces and structures", while we now outline a possible classification of spaces (and structures) in the spirit of Bourbaki.


title: Space (mathematics)

text: We classify spaces on three levels. Given that each mathematical theory describes its objects by some of their properties, the first question to ask is: which properties? This leads to the first (upper) classification level. On the second level, one takes into account answers to especially important questions (among the questions that make sense according to the first level). On the third level of classification, one takes into account answers to all possible questions.


title: Space (mathematics)

text: For example, the "upper-level classification" distinguishes between Euclidean and projective spaces, since the distance between two points is defined in Euclidean spaces but undefined in projective spaces.


title: Space (mathematics)

text: Another example. The question "what is the sum of the three angles of a triangle" makes sense in a Euclidean space but not in a projective space. In a non-Euclidean space the question makes sense but is answered differently, which is not an upper-level distinction.


title: Space (mathematics)

text: Also, the distinction between a Euclidean plane and a Euclidean 3-dimensional space is not an upper-level distinction; the question "what is the dimension" makes sense in both cases.


title: Space (mathematics)

text: The "second-level classification" distinguishes, for example, between Euclidean and non-Euclidean spaces; between finite-dimensional and infinite-dimensional spaces; between compact and non-compact spaces, etc.


title: Space (mathematics)

text: In Bourbaki's terms, the second-level classification is the classification by "species". Unlike biological taxonomy, a space may belong to several species.


title: Space (mathematics)

text: The "third-level classification" distinguishes, for example, between spaces of different dimension, but does not distinguish between a plane of a three-dimensional Euclidean space, treated as a two-dimensional Euclidean space, and the set of all pairs of real numbers, also treated as a two-dimensional Euclidean space. Likewise it does not distinguish between different Euclidean models of the same non-Euclidean space.


title: Space (mathematics)

text: More formally, the third level classifies spaces up to isomorphism. An isomorphism between two spaces is defined as a one-to-one correspondence between the points of the first space and the points of the second space, that preserves all relations stipulated according to the first level. Mutually isomorphic spaces are thought of as copies of a single space. If one of them belongs to a given species then they all do.


title: Space (mathematics)

text: The notion of isomorphism sheds light on the upper-level classification. Given a one-to-one correspondence between two spaces of the same upper-level class, one may ask whether it is an isomorphism or not. This question makes no sense for two spaces of different classes.


title: Space (mathematics)

text: An isomorphism to itself is called an automorphism. Automorphisms of a Euclidean space are shifts, rotations, reflections and compositions of these. Euclidean space is homogeneous in the sense that every point can be transformed into every other point by some automorphism.


title: Space (mathematics)

text: Euclidean axioms leave no freedom; they determine uniquely all geometric properties of the space. More exactly: all three-dimensional Euclidean spaces are mutually isomorphic. In this sense we have "the" three-dimensional Euclidean space. In Bourbaki's terms, the corresponding theory is "univalent". In contrast, topological spaces are generally non-isomorphic; their theory is "multivalent". A similar idea occurs in mathematical logic: a theory is called categorical if all its models of the same cardinality are mutually isomorphic. According to Bourbaki, the study of multivalent theories is the most striking feature which distinguishes modern mathematics from classical mathematics.


title: Space (mathematics)

text: Topological notions (continuity, convergence, open sets, closed sets etc.) are defined naturally in every Euclidean space. In other words, every Euclidean space is also a topological space. Every isomorphism between two Euclidean spaces is also an isomorphism between the corresponding topological spaces (called "homeomorphism"), but the converse is wrong: a homeomorphism may distort distances. In Bourbaki's terms, "topological space" is an "underlying" structure of the "Euclidean space" structure. Similar ideas occur in category theory: the category of Euclidean spaces is a concrete category over the category of topological spaces; the forgetful (or "stripping") functor maps the former category to the latter category.


title: Space (mathematics)

text: A three-dimensional Euclidean space is a special case of a Euclidean space. In Bourbaki's terms, the species of three-dimensional Euclidean space is "richer" than the species of Euclidean space. Likewise, the species of compact topological space is richer than the species of topological space.


title: Space (mathematics)

text: Such relations between species of spaces may be expressed diagrammatically as shown in Fig. 3. An arrow from A to B means that every is also a or may be treated as a or provides a etc. Treating A and B as classes of spaces one may interpret the arrow as a transition from A to B. (In Bourbaki's terms, "procedure of deduction" of a from a Not quite a function unless the classes A,B are sets; this nuance does not invalidate the following.) The two arrows on Fig. 3 are not invertible, but for different reasons.


title: Space (mathematics)

text: The transition from "Euclidean" to "topological" is forgetful. Topology distinguishes continuous from discontinuous, but does not distinguish rectilinear from curvilinear. Intuition tells us that the Euclidean structure cannot be restored from the topology. A proof uses an automorphism of the topological space (that is, self-homeomorphism) that is not an automorphism of the Euclidean space (that is, not a composition of shifts, rotations and reflections). Such transformation turns the given Euclidean structure into a (isomorphic but) different Euclidean structure; both Euclidean structures correspond to a single topological structure.


title: Space (mathematics)

text: In contrast, the transition from "3-dim Euclidean" to "Euclidean" is not forgetful; a Euclidean space need not be 3-dimensional, but if it happens to be 3-dimensional, it is full-fledged, no structure is lost. In other words, the latter transition is injective (one-to-one), while the former transition is not injective (many-to-one). We denote injective transitions by an arrow with a barbed tail, "↣" rather than "→".


title: Space (mathematics)

text: Both transitions are not surjective, that is, not every B-space results from some A-space. First, a 3-dim Euclidean space is a special (not general) case of a Euclidean space. Second, a topology of a Euclidean space is a special case of topology (for instance, it must be non-compact, and connected, etc). We denote surjective transitions by a two-headed arrow, "↠" rather than "→". See for example Fig. 4; there, the arrow from "real linear topological" to "real linear" is two-headed, since every real linear space admits some (at least one) topology compatible with its linear structure.


title: Space (mathematics)

text: Such topology is non-unique in general, but unique when the real linear space is finite-dimensional. For these spaces the transition is both injective and surjective, that is, bijective; see the arrow from "finite-dim real linear topological" to "finite-dim real linear" on Fig. 4. The inverse transition exists (and could be shown by a second, backward arrow). The two species of structures are thus equivalent. In practice, one makes no distinction between equivalent species of structures. Equivalent structures may be treated as a single structure, as shown by a large box on Fig. 4.


title: Space (mathematics)

text: The transitions denoted by the arrows obey isomorphisms. That is, two isomorphic lead to two isomorphic .


title: Space (mathematics)

text: The diagram on Fig. 4 is commutative. That is, all directed paths in the diagram with the same start and endpoints lead to the same result. Other diagrams below are also commutative, except for dashed arrows on Fig. 9. The arrow from "topological" to "measurable" is dashed for the reason explained there: "In order to turn a topological space into a measurable space one endows it with a σ-algebra. The σ-algebra of Borel sets is the most popular, but not the only choice." A solid arrow denotes a prevalent, so-called "canonical" transition that suggests itself naturally and is widely used, often implicitly, by default. For example, speaking about a continuous function on a Euclidean space, one need not specify its topology explicitly. In fact, alternative topologies exist and are used sometimes, for example, the fine topology; but these are always specified explicitly, since they are much less notable that the prevalent topology. A dashed arrow indicates that several transitions are in use and no one is quite prevalent.


title: Space (mathematics)

text: Linear spaces are of algebraic nature; there are real linear spaces (over the field of real numbers),


title: Space (mathematics)

text: complex linear spaces (over the field of complex numbers), and more generally, linear spaces over any field. Every complex linear space is also a real linear space (the latter "underlies" the former), since each complex number can be specified by two real numbers. For example, the complex plane treated as a one-dimensional complex linear space may be downgraded to a two-dimensional real linear space. In contrast, the real line can be treated as a one-dimensional real linear space but not a complex linear space. See also field extensions. More generally, a vector space over a field also has the structure of a vector space over a subfield of that field.


title: Space (mathematics)

text: Linear operations, given in a linear space by definition, lead to such notions as straight lines (and planes, and other linear subspaces); parallel lines; ellipses (and ellipsoids). However, it is impossible to define


title: Space (mathematics)

text: there is no structure like a scalar product that could be used for measuring angles. The dimension of a linear space is defined as the maximal number of linearly independent vectors or, equivalently, as the minimal number of vectors that span the space; it may be finite or infinite. Two linear spaces over the same field are isomorphic if and only if they are of the same dimension. A complex linear space is also a real linear space.


title: Space (mathematics)

text: Topological spaces are of analytic nature. Open sets, given in a topological space by definition, lead to such notions as continuous functions, paths, maps; convergent sequences, limits; interior, boundary, exterior. However, uniform continuity, bounded sets, Cauchy sequences, differentiable functions (paths, maps) remain undefined. Isomorphisms between topological spaces are traditionally called homeomorphisms; these are one-to-one correspondences continuous in both directions. The open interval (0,1) is homeomorphic to the whole real line (−∞,∞) but not homeomorphic to the closed interval [0,1], nor to a circle. The surface of a cube is homeomorphic to a sphere (the surface of a ball) but not homeomorphic to a torus. Euclidean spaces of different dimensions are not homeomorphic, which seems evident, but is not easy to prove. The dimension of a topological space is difficult to define; inductive dimension (based on the observation that the dimension of the boundary of a geometric figure is usually one less than the dimension of the figure itself) and Lebesgue covering dimension can be used. In the case of a Euclidean space, both topological dimensions are equal to "n".


title: Space (mathematics)

text: Every subset of a topological space is itself a topological space (in contrast, only "linear" subsets of a linear space are linear spaces). Arbitrary topological spaces, investigated by general topology (called also point-set topology) are too diverse for a complete classification up to homeomorphism. Compact topological spaces are an important class of topological spaces ("species" of this "type"). Every continuous function is bounded on such space. The closed interval [0,1] and the extended real line [−∞,∞] are compact; the open interval (0,1) and the line (−∞,∞) are not. Geometric topology investigates manifolds (another "species" of this "type"); these are topological spaces locally homeomorphic to Euclidean spaces (and satisfying a few extra conditions). Low-dimensional manifolds are completely classified up to homeomorphism.


title: Space (mathematics)

text: Both the linear and topological structures underlie the linear topological space (in other words, topological vector space) structure. A linear topological space is both a real or complex linear space and a topological space, such that the linear operations are continuous. So a linear space that is also topological is not in general a linear topological space.


title: Space (mathematics)

text: Every finite-dimensional real or complex linear space is a linear topological space in the sense that it carries one and only one topology that makes it a linear topological space. The two structures, "finite-dimensional real or complex linear space" and "finite-dimensional linear topological space", are thus equivalent, that is, mutually underlying. Accordingly, every invertible linear transformation of a finite-dimensional linear topological space is a homeomorphism. The three notions of dimension (one algebraic and two topological) agree for finite-dimensional real linear spaces. In infinite-dimensional spaces, however, different topologies can conform to a given linear structure, and invertible linear transformations are generally not homeomorphisms.


title: Space (mathematics)

text: It is convenient to introduce affine and projective spaces by means of linear spaces, as follows. A linear subspace of a linear space, being itself a linear space, is not homogeneous; it contains a special point, the origin. Shifting it by a vector external to it, one obtains a affine subspace. It is homogeneous. An affine space need not be included into a linear space, but is isomorphic to an affine subspace of a linear space. All affine spaces are mutually isomorphic. In the words of John Baez, "an affine space is a vector space that's forgotten its origin". In particular, every linear space is also an affine space.


title: Space (mathematics)

text: Given an affine subspace "A" in a linear space "L", a straight line in "A" may be defined as the intersection of "A" with a linear subspace of "L" that intersects "A": in other words, with a plane through the origin that is not parallel to "A". More generally, a affine subspace of "A" is the intersection of "A" with a linear subspace of "L" that intersects "A".


title: Space (mathematics)

text: Every point of the affine subspace "A" is the intersection of "A" with a linear subspace of "L". However, some subspaces of "L" are parallel to "A"; in some sense, they intersect "A" at infinity. The set of all linear subspaces of a linear space is, by definition, a projective space. And the affine subspace "A" is embedded into the projective space as a proper subset. However, the projective space itself is homogeneous. A straight line in the projective space corresponds to a linear subspace of the ("n"+1)-dimensional linear space. More generally, a projective subspace of the projective space corresponds to a linear subspace of the ("n"+1)-dimensional linear space, and is isomorphic to the projective space.


title: Space (mathematics)

text: Defined this way, affine and projective spaces are of algebraic nature; they can be real, complex, and more generally, over any field.


title: Space (mathematics)

text: Every real or complex affine or projective space is also a topological space. An affine space is a non-compact manifold; a projective space is a compact manifold. In a real projective space a straight line is homeomorphic to a circle, therefore compact, in contrast to a straight line in a linear of affine space.


title: Space (mathematics)

text: Distances between points are defined in a metric space. Isomorphisms between metric spaces are called isometries. Every metric space is also a topological space. A topological space is called metrizable, if it underlies a metric space. All manifolds are metrizable.


title: Space (mathematics)

text: bounded sets and Cauchy sequences. A metric space is called complete if all Cauchy sequences converge. Every incomplete space is isometrically embedded, as a dense subset, into a complete space (the completion). Every compact metric space is complete; the real line is non-compact but complete; the open interval (0,1) is incomplete.


title: Space (mathematics)

text: Every Euclidean space is also a complete metric space. Moreover, all geometric notions immanent to a Euclidean space can be characterized in terms of its metric. For example, the straight segment connecting two given points "A" and "C" consists of all points "B" such that the distance between "A" and "C" is equal to the sum of two distances, between "A" and "B" and between "B" and "C".


title: Space (mathematics)

text: The Hausdorff dimension (related to the number of small balls that cover the given set) applies to metric spaces, and can be non-integer (especially for fractals). For a Euclidean space, the Hausdorff dimension is equal to "n".


title: Space (mathematics)

text: Uniform spaces do not introduce distances, but still allow one to use uniform continuity, Cauchy sequences (or filters or nets), completeness and completion. Every uniform space is also a topological space. Every "linear" topological space (metrizable or not) is also a uniform space, and is complete in finite dimension but generally incomplete in infinite dimension. More generally, every commutative topological group is also a uniform space. A non-commutative topological group, however, carries two uniform structures, one left-invariant, the other right-invariant.


title: Space (mathematics)

text: Vectors in a Euclidean space form a linear space, but each vector formula_1 has also a length, in other words, norm, formula_2. A real or complex linear space endowed with a norm is a normed space. Every normed space is both a linear topological space and a metric space. A Banach space is a complete normed space. Many spaces of sequences or functions are infinite-dimensional Banach spaces.


title: Space (mathematics)

text: The set of all vectors of norm less than one is called the unit ball of a normed space. It is a convex, centrally symmetric set, generally not an ellipsoid; for example, it may be a polygon (in the plane) or, more generally, a polytope (in arbitrary finite dimension). The parallelogram law (called also parallelogram identity)


title: Space (mathematics)

text: generally fails in normed spaces, but holds for vectors in Euclidean spaces, which follows from the fact that the squared Euclidean norm of a vector is its inner product with itself, formula_4.


title: Space (mathematics)

text: An inner product space is a real or complex linear space, endowed with a bilinear or respectively sesquilinear form, satisfying some conditions and called an inner product. Every inner product space is also a normed space. A normed space underlies an inner product space if and only if it satisfies the parallelogram law, or equivalently, if its unit ball is an ellipsoid. Angles between vectors are defined in inner product spaces. A Hilbert space is defined as a complete inner product space. (Some authors insist that it must be complex, others admit also real Hilbert spaces.) Many spaces of sequences or functions are infinite-dimensional Hilbert spaces. Hilbert spaces are very important for quantum theory.


title: Space (mathematics)

text: All real inner product spaces are mutually isomorphic. One may say that the Euclidean space is the real inner product space that forgot its origin.


title: Space (mathematics)

text: Smooth manifolds are not called "spaces", but could be. Every smooth manifold is a topological manifold, and can be embedded into a finite-dimensional linear space. Smooth surfaces in a finite-dimensional linear space are smooth manifolds: for example, the surface of an ellipsoid is a smooth manifold, a polytope is not. Real or complex finite-dimensional linear, affine and projective spaces are also smooth manifolds.


title: Space (mathematics)

text: At each one of its points, a smooth path in a smooth manifold has a tangent vector that belongs to the manifold's tangent space at this point. Tangent spaces to an smooth manifold are linear spaces. The differential of a smooth function on a smooth manifold provides a linear functional on the tangent space at each point.


title: Space (mathematics)

text: A Riemannian manifold, or Riemann space, is a smooth manifold whose tangent spaces are endowed with inner products satisfying some conditions. Euclidean spaces are also Riemann spaces. Smooth surfaces in Euclidean spaces are Riemann spaces. A hyperbolic space is also a Riemann space. A curve in a Riemann space has a length, and the length of the shortest curve between two points defines a distance, such that the Riemann space is a metric space. The angle between two curves intersecting at a point is the angle between their tangent lines.


title: Space (mathematics)

text: Waiving positivity of inner products on tangent spaces, one obtains pseudo-Riemann spaces, including the Lorentzian spaces that are very important for general relativity.


title: Space (mathematics)

text: Waiving distances and angles while retaining volumes (of geometric bodies) one reaches measure theory. Besides the volume, a measure generalizes the notions of area, length, mass (or charge) distribution, and also probability distribution, according to Andrey Kolmogorov's approach to probability theory.


title: Space (mathematics)

text: A "geometric body" of classical mathematics is much more regular than just a set of points. The boundary of the body is of zero volume. Thus, the volume of the body is the volume of its interior, and the interior can be exhausted by an infinite sequence of cubes. In contrast, the boundary of an arbitrary set of points can be of non-zero volume (an example: the set of all rational points inside a given cube). Measure theory succeeded in extending the notion of volume to a vast class of sets, the so-called measurable sets. Indeed, non-measurable sets almost never occur in applications.


title: Space (mathematics)

text: Measurable sets, given in a measurable space by definition, lead to measurable functions and maps. In order to turn a topological space into a measurable space one endows it with a The of Borel sets is the most popular, but not the only choice. (Baire sets, universally measurable sets, etc, are also used sometimes.)


title: Space (mathematics)

text: The topology is not uniquely determined by the Borel for example, the norm topology and the weak topology on a separable Hilbert space lead to the same Borel .


title: Space (mathematics)

text: Actually, a can be generated by a given collection of sets (or functions) irrespective of any topology. Every subset of a measurable space is itself a measurable space.


title: Space (mathematics)

text: Standard measurable spaces (also called standard Borel spaces) are especially useful due to some similarity to compact spaces (see EoM). Every bijective measurable mapping between standard measurable spaces is an isomorphism; that is, the inverse mapping is also measurable. And a mapping between such spaces is measurable if and only if its graph is measurable in the product space. Similarly, every bijective continuous mapping between compact metric spaces is a homeomorphism; that is, the inverse mapping is also continuous. And a mapping between such spaces is continuous if and only if its graph is closed in the product space.


title: Space (mathematics)

text: Every Borel set in a Euclidean space (and more generally, in a complete separable metric space), endowed with the Borel is a standard measurable space. All uncountable standard measurable spaces are mutually isomorphic.


title: Space (mathematics)

text: A measure space is a measurable space endowed with a measure. A Euclidean space with the Lebesgue measure is a measure space. Integration theory defines integrability and integrals of measurable functions on a measure space.


title: Space (mathematics)

text: Sets of measure 0, called null sets, are negligible. Accordingly, a "mod 0 isomorphism" is defined as isomorphism between subsets of full measure (that is, with negligible complement).


title: Space (mathematics)

text: A probability space is a measure space such that the measure of the whole space is equal to 1. The product of any family (finite or not) of probability spaces is a probability space. In contrast, for measure spaces in general, only the product of finitely many spaces is defined. Accordingly, there are many infinite-dimensional probability measures (especially, Gaussian measures), but no infinite-dimensional Lebesgue measures.


title: Space (mathematics)

text: Standard probability spaces are especially useful. On a standard probability space a conditional expectation may be treated as the integral over the conditional measure (regular conditional probabilities, see also disintegration of measure). Given two standard probability spaces, every homomorphism of their measure algebras is induced by some measure preserving map. Every probability measure on a standard measurable space leads to a standard probability space. The product of a sequence (finite or not) of standard probability spaces is a standard probability space. All non-atomic standard probability spaces are mutually isomorphic mod 0; one of them is the interval (0,1) with the Lebesgue measure.


title: Space (mathematics)

text: These spaces are less geometric. In particular, the idea of dimension, applicable (in one form or another) to all other spaces, does not apply to measurable, measure and probability spaces.


title: Space (mathematics)

text: The theoretical study of calculus, known as mathematical analysis, led in the early 20th century to the consideration of linear spaces of real-valued or complex-valued functions. The earliest examples of these were function spaces, each one adapted to its own class of problems. These examples shared many common features, and these features were soon abstracted into Hilbert spaces, Banach spaces, and more general topological vector spaces. These were a powerful toolkit for the solution of a wide range of mathematical problems.


title: Space (mathematics)

text: The most detailed information was carried by a class of spaces called Banach algebras. These are Banach spaces together with a continuous multiplication operation. An important early example was the Banach algebra of essentially bounded measurable functions on a measure space "X". This set of functions is a Banach space under pointwise addition and scalar multiplication. With the operation of pointwise multiplication, it becomes a special type of Banach space, one now called a commutative von Neumann algebra. Pointwise multiplication determines a representation of this algebra on the Hilbert space of square integrable functions on "X". An early observation of John von Neumann was that this correspondence also worked in reverse: Given some mild technical hypotheses, a commutative von Neumann algebra together with a representation on a Hilbert space determines a measure space, and these two constructions (of a von Neumann algebra plus a representation and of a measure space) are mutually inverse.


title: Space (mathematics)

text: Von Neumann then proposed that non-commutative von Neumann algebras should have geometric meaning, just as commutative von Neumann algebras do. Together with Francis Murray, he produced a classification of von Neumann algebras. The direct integral construction shows how to break any von Neumann algebra into a collection of simpler algebras called "factors". Von Neumann and Murray classified factors into three types. Type I was nearly identical to the commutative case. Types II and III exhibited new phenomena. A type II von Neumann algebra determined a geometry with the peculiar feature that the dimension could be any non-negative real number, not just an integer. Type III algebras were those that were neither types I nor II, and after several decades of effort, these were proven to be closely related to type II factors.


title: Space (mathematics)

text: A slightly different approach to the geometry of function spaces developed at the same time as von Neumann and Murray's work on the classification of factors. This approach is the theory of Here, the motivating example is the formula_5, where "X" is a locally compact Hausdorff topological space. By definition, this is the algebra of continuous complex-valued functions on "X" that vanish at infinity (which loosely means that the farther you go from a chosen point, the closer the function gets to zero) with the operations of pointwise addition and multiplication. The Gelfand–Naimark theorem implied that there is a correspondence between commutative and geometric objects: Every commutative is of the form formula_5 for some locally compact Hausdorff space "X". Consequently it is possible to study locally compact Hausdorff spaces purely in terms of commutative Non-commutative geometry takes this as inspiration for the study of non-commutative If there were such a thing as a "non-commutative space "X"," then its formula_5 would be a non-commutative ; if in addition the Gelfand–Naimark theorem applied to these non-existent objects, then spaces (commutative or not) would be the same as so, for lack of a direct approach to the definition of a non-commutative space, a non-commutative space is "defined" to be a non-commutative Many standard geometric tools can be restated in terms of and this gives geometrically-inspired techniques for studying non-commutative .


title: Space (mathematics)

text: Both of these examples are now cases of a field called non-commutative geometry. The specific examples of von Neumann algebras and are known as non-commutative measure theory and non-commutative topology, respectively. Non-commutative geometry is not merely a pursuit of generality for its own sake and is not just a curiosity. Non-commutative spaces arise naturally, even inevitably, from some constructions. For example, consider the non-periodic Penrose tilings of the plane by kites and darts. It is a theorem that, in such a tiling, every finite patch of kites and darts appears infinitely often. As a consequence, there is no way to distinguish two Penrose tilings by looking at a finite portion. This makes it impossible to assign the set of all tilings a topology in the traditional sense. Despite this, the Penrose tilings determine a non-commutative and consequently they can be studied by the techniques of non-commutative geometry. Another example, and one of great interest within differential geometry, comes from foliations of manifolds. These are ways of splitting the manifold up into smaller-dimensional submanifolds called "leaves", each of which is locally parallel to others nearby. The set of all leaves can be made into a topological space. However, the example of an irrational rotation shows that this topological space can be inaccessible to the techniques of classical measure theory. However, there is a non-commutative von Neumann algebra associated to the leaf space of a foliation, and once again, this gives an otherwise unintelligible space a good geometric structure.


title: Space (mathematics)

text: Algebraic geometry studies the geometric properties of polynomial equations. Polynomials are a type of function defined from the basic arithmetic operations of addition and multiplication. Because of this, they are closely tied to algebra. Algebraic geometry offers a way to apply geometric techniques to questions of pure algebra, and vice versa.


title: Space (mathematics)

text: Prior to the 1940s, algebraic geometry worked exclusively over the complex numbers, and the most fundamental variety was projective space. The geometry of projective space is closely related to the theory of perspective, and its algebra is described by homogeneous polynomials. All other varieties were defined as subsets of projective space. Projective varieties were subsets defined by a set of homogeneous polynomials. At each point of the projective variety, all the polynomials in the set were required to equal zero. The complement of the zero set of a linear polynomial is an affine space, and an affine variety was the intersection of a projective variety with an affine space.


title: Space (mathematics)

text: André Weil saw that geometric reasoning could sometimes be applied in number-theoretic situations where the spaces in question might be discrete or even finite. In pursuit of this idea, Weil rewrote the foundations of algebraic geometry, both freeing algebraic geometry from its reliance on complex numbers and introducing "abstract algebraic varieties" which were not embedded in projective space. These are now simply called "varieties".


title: Space (mathematics)

text: The type of space that underlies most modern algebraic geometry is even more general than Weil's abstract algebraic varieties. It was introduced by Alexander Grothendieck and is called a scheme. One of the motivations for scheme theory is that polynomials are unusually structured among functions, and algebraic varieties are consequently rigid. This presents problems when attempting to study degenerate situations. For example, almost any pair of points on a circle determines a unique line called the secant line, and as the two points move around the circle, the secant line varies continuously. However, when the two points collide, the secant line degenerates to a tangent line. The tangent line is unique, but the geometry of this configuration—a single point on a circle—is not expressive enough to determine a unique line. Studying situations like this requires a theory capable of assigning extra data to degenerate situations.


title: Space (mathematics)

text: One of the building blocks of a scheme is a topological space. Topological spaces have continuous functions, but continuous functions are too general to reflect the underlying algebraic structure of interest. The other ingredient in a scheme, therefore, is a sheaf on the topological space, called the "structure sheaf". On each open subset of the topological space, the sheaf specifies a collection of functions, called "regular functions". The topological space and the structure sheaf together are required to satisfy conditions that mean the functions come from algebraic operations.


title: Space (mathematics)

text: Like manifolds, schemes are defined as spaces that are locally modeled on a familiar space. In the case of manifolds, the familiar space is Euclidean space. For a scheme, the local models are called affine schemes. Affine schemes provide a direct link between algebraic geometry and commutative algebra. The fundamental objects of study in commutative algebra are commutative rings. If formula_8 is a commutative ring, then there is a corresponding affine scheme formula_9 which translates the algebraic structure of formula_8 into geometry. Conversely, every affine scheme determines a commutative ring, namely, the ring of global sections of its structure sheaf. These two operations are mutually inverse, so affine schemes provide a new language with which to study questions in commutative algebra. By definition, every point in a scheme has an open neighborhood which is an affine scheme.


title: Space (mathematics)

text: There are many schemes that are not affine. In particular, projective spaces satisfy a condition called properness which is analogous to compactness. Affine schemes cannot be proper (except in trivial situations like when the scheme has only a single point), and hence no projective space is an affine scheme (except for zero-dimensional projective spaces). Projective schemes, meaning those that arise as closed subschemes of a projective space, are the single most important family of schemes.


title: Space (mathematics)

text: Several generalizations of schemes have been introduced. Michael Artin defined an algebraic space as the quotient of a scheme by the equivalence relations that define étale morphisms. Algebraic spaces retain many of the useful properties of schemes while simultaneously being more flexible. For instance, the Keel–Mori theorem can be used to show that many moduli spaces are algebraic spaces.


title: Space (mathematics)

text: More general than an algebraic space is a Deligne–Mumford stack. DM stacks are similar to schemes, but they permit singularities that cannot be described solely in terms of polynomials. They play the same role for schemes that orbifolds do for manifolds. For example, the quotient of the affine plane by a finite group of rotations around the origin yields a Deligne–Mumford stack that is not a scheme or an algebraic space. Away from the origin, the quotient by the group action identifies finite sets of equally spaced points on a circle. But at the origin, the circle consists of only a single point, the origin itself, and the group action fixes this point. In the quotient DM stack, however, this point comes with the extra data of being a quotient. This kind of refined structure is useful in the theory of moduli spaces, and in fact, it was originally introduced to describe moduli of algebraic curves.


title: Space (mathematics)

text: A further generalization are the algebraic stacks, also called Artin stacks. DM stacks are limited to quotients by finite group actions. While this suffices for many problems in moduli theory, it is too restrictive for others, and Artin stacks permit more general quotients.


title: Space (mathematics)

text: In Grothendieck's work on the Weil conjectures, he introduced a new type of topology now called a Grothendieck topology. A topological space (in the ordinary sense) axiomatizes the notion of "nearness," making two points be nearby if and only if they lie in many of the same open sets. By contrast, a Grothendieck topology axiomatizes the notion of "covering". A covering of a space is a collection of subspaces that jointly contain all the information of the ambient space. Since sheaves are defined in terms of coverings, a Grothendieck topology can also be seen as an axiomatization of the theory of sheaves.


title: Space (mathematics)

text: Grothendieck's work on his topologies led him to the theory of topoi. In his memoir "Récoltes et Semailles", he called them his "most vast conception". A sheaf (either on a topological space or with respect to a Grothendieck topology) is used to express local data. The category of all sheaves carries all possible ways of expressing local data. Since topological spaces are constructed from points, which are themselves a kind of local data, the category of sheaves can therefore be used as a replacement for the original space. Grothendieck consequently defined a topos to be a category of sheaves and studied topoi as objects of interest in their own right. These are now called Grothendieck topoi.


title: Space (mathematics)

text: Every topological space determines a topos, and vice versa. There are topological spaces where taking the associated topos loses information, but these are generally considered pathological. (A necessary and sufficient condition is that the topological space be a sober space.) Conversely, there are topoi whose associated topological spaces do not capture the original topos. But, far from being pathological, these topoi can be of great mathematical interest. For instance, Grothendieck's theory of étale cohomology (which eventually led to the proof of the Weil conjectures) can be phrased as cohomology in the étale topos of a scheme, and this topos does not come from a topological space.


title: Space (mathematics)

text: Topological spaces in fact lead to very special topoi called locales. The set of open subsets of a topological space determines a lattice. The axioms for a topological space cause these lattices to be complete Heyting algebras. The theory of locales takes this as its starting point. A locale is defined to be a complete Heyting algebra, and the elementary properties of topological spaces are re-expressed and reproved in these terms. The concept of a locale turns out to be more general than a topological space, in that every sober topological space determines a unique locale, but many interesting locales do not come from topological spaces. Because locales need not have points, the study of locales is somewhat jokingly called pointless topology.


title: Space (mathematics)

text: Topoi also display deep connections to mathematical logic. Every Grothendieck topos has a special sheaf called a subobject classifier. This subobject classifier functions like the set of all possible truth values. In the topos of sets, the subobject classifier is the set formula_11, corresponding to "False" and "True". But in other topoi, the subobject classifier can be much more complicated. Lawvere and Tierney recognized that axiomatizing the subobject classifier yielded a more general kind of topos, now known as an elementary topos, and that elementary topoi were models of intuitionistic logic. In addition to providing a powerful way to apply tools from logic to geometry, this made possible the use of geometric methods in logic.


title: Space (mathematics)

text: Nevertheless, a general definition of "structure" was proposed by Bourbaki; it embraces all types of spaces mentioned above, (nearly?) all types of mathematical structures used till now, and more. It provides a general definition of isomorphism, and justifies transfer of properties between isomorphic structures. However, it was never used actively in mathematical practice (not even in the mathematical treatises written by Bourbaki himself). Here are the last phrases from a review by Robert Reed of a book by Leo Corry:


title: Space (mathematics)

text: For more information on mathematical structures see Wikipedia: mathematical structure, equivalent definitions of mathematical structures, and transport of structure.


title: Space (mathematics)

text: The distinction between geometric "spaces" and algebraic "structures" is sometimes clear, sometimes elusive. Clearly, groups are algebraic, while Euclidean spaces are geometric. Modules over rings are as algebraic as groups. In particular, when the ring appears to be a field, the module appears to be a linear space; is it algebraic or geometric? In particular, when it is finite-dimensional, over real numbers, and endowed with inner product, it becomes Euclidean space; now geometric. The (algebraic?) field of real numbers is the same as the (geometric?) real line. Its algebraic closure, the (algebraic?) field of complex numbers, is the same as the (geometric?) complex plane. It is first of all "a place we do analysis" (rather than algebra or geometry).


title: Space (mathematics)

text: Every space treated in Section "Types of spaces" above, except for "Non-commutative geometry", "Schemes" and "Topoi" subsections, is a set (the "principal base set" of the structure, according to Bourbaki) endowed with some additional structure; elements of the base set are usually called "points" of this space. In contrast, elements of (the base set of) an algebraic structure usually are not called "points".


title: Space (mathematics)

text: However, sometimes one uses more than one principal base set. For example, two-dimensional projective geometry may be formalized via two base sets, the set of points and the set of lines. Moreover, a striking feature of projective planes is the symmetry of the roles played by points and lines. A less geometric example: a graph may be formalized via two base sets, the set of vertices (called also nodes or points) and the set of edges (called also arcs or lines). Generally, finitely many principal base sets and finitely many auxiliary base sets are stipulated by Bourbaki.


title: Space (mathematics)

text: Many mathematical structures of geometric flavor treated in the "Non-commutative geometry", "Schemes" and "Topoi" subsections above do not stipulate a base set of points. For example, "pointless topology" (in other words, point-free topology, or locale theory) starts with a single base set whose elements imitate open sets in a topological space (but are not sets of points); see also mereotopology and point-free geometry.


title: Return to Me

text: Return to Me is a 2000 American romantic comedy-drama film directed by Bonnie Hunt and starring David Duchovny and Minnie Driver. It was filmed in Chicago and was released on April 7, 2000 by Metro-Goldwyn-Mayer. It was Carroll O'Connor's final film before his death the following year.


title: Return to Me

text: Bob and Elizabeth Rueland (David Duchovny and Joely Richardson) live and work in Chicago — Bob as an architect, Elizabeth as a zoologist at Lincoln Park Zoo.


title: Return to Me

text: On the night of her fundraiser for a new primate house, Bob promises Elizabeth he'll finish the building. Elizabeth is killed in a car accident leaving the fundraiser, and her heart is transplanted to artist Grace Briggs (Minnie Driver), who has suffered from heart disease since the age of 14 and is near death.


title: Return to Me

text: The surgery is successful; Grace is able to live a normal life for the first time and plans to make her first airplane trip to Italy to paint. Grace's best friend Megan Dayton (Bonnie Hunt), encourages her to start dating in spite of her self-consciousness about the long surgical scar on her chest.


title: Return to Me

text: Grace writes a letter to the donor's family after the surgery, thanking them for the heart she received; it takes her over a year to finally find the courage to mail the letter. Bob works to build the primate house Elizabeth raised money for; he is still depressed a year after her death. He recognizes that he must resume his life as he becomes frustrated seeing that his dog hasn't gotten over the loss either.


title: Return to Me

text: Bob's friend, veterinarian Charlie (David Alan Grier), organizes a blind date for him at O'Reilly's, a self-styled Irish-Italian restaurant. The date goes very badly, as his date is very obnoxious, petty and self-absorbed. However, Bob finds that he is drawn to the waitress — Grace, who is also the granddaughter of the restaurant's owner, Marty O'Reilly (Carroll O'Connor).


title: Return to Me

text: Although they are both unaware of the connection they have through Elizabeth's heart, Bob and Grace begin to date. As they grow closer together, she is reluctant to tell him about her medical history. After several months of dating, Grace finally decides to tell Bob about her transplant. However, before she gets the chance, she finds in his house the letter she had sent several months earlier.


title: Return to Me

text: Horrified by the discovery, Grace flees and tells Megan what has happened. Megan's husband, Joe, (Jim Belushi) becomes infuriated as he has misunderstood Grace's panic and thinks Bob must be married. Megan then explains the situation to him in six monosyllables: "Grace has Bob's dead wife's heart!" When Grace meets Bob again, she tells him the truth. Speechless, he leaves.


title: Return to Me

text: Against Megan's advice to not run away from the situation, Grace goes to Italy alone. Back at home, Bob realizes that although he will always miss Elizabeth, he "aches" for Grace. He decides to go after her, and they reunite in Italy. They return to Chicago for the dedication of the new primate house.


title: Return to Me

text: As the film ends we see Wally (William Bronder) and Sophie (Marianne Muellerleile) dancing happily at their wedding reception at O'Reilly's restaurant, and Charlie holding Joe and a visibly pregnant Megan's toddler, while they join Grace and Bob on the dance floor.


title: Return to Me

text: The film received generally mixed reviews. Based on 99 reviews, it has a score of 62% on Rotten Tomatoes with the consensus reading "David Duchovny and Minnie Driver provide heart-warming romance and comedy in this solid debut by director Bonnie Hunt." Peter Stack of the "San Francisco Chronicle" stated, "Old-fashioned as all get-out, "Return to Me" is swathed in an unabashed feel-good tone." Roger Ebert called it "so innocent, so naive, so sweet and sincere, that you must leave your cynicism at the door or choose another movie.""Entertainment Weekly"'s Lisa Schwarzbaum gave it a C+ grade, and stated "the alluringly deadpan Duchovny can make no headway with Driver." Jay Carr of "The Boston Globe" called it "ultimately too bland and safe."


title: Return to Me

text: The film opened at fourth place at the North American box office making US$7.8 million in its opening weekend, behind "The Road to El Dorado", "Erin Brockovich" and "Rules of Engagement", which opened at the top spot. It would make a total of $32,662,299 in its entire box office run.


title: Monongahela River

text: The Monongahela River ( , )—often referred to locally as the Mon ()—is a river on the Allegheny Plateau in north-central West Virginia and Southwestern Pennsylvania. The river flows from the confluence of its west and east forks in north-central West Virginia northeasterly into southwestern Pennsylvania, then northerly to Pittsburgh and its confluence with the Allegheny River to form the Ohio River. The river's entire length is navigable via a series of locks and dams.


title: Monongahela River

text: The Unami word "Monongahela" means "falling banks", in reference to the geological instability of the river's banks. Moravian missionary David Zeisberger (1721–1808) gave this account of the naming: "In the Indian tongue the name of this river was "Mechmenawungihilla" (alternatively spelled "Menawngihella"), which signifies a high bank, which is ever washed out and therefore collapses."


title: Monongahela River

text: The Lenape Language Project renders the word as "Mënaonkihëla" (pronounced ), translated "where banks cave in or erode", from the verbs "mënaonkihële" "the dirt caves off" (such as the bank of a river or creek, or in a landslide) and "mënaonke" (pronounced ), "it has a loose bank" (where one might fall in).


title: Monongahela River

text: Monongalia County and the town of Monongah, both in West Virginia, are named for the river, as is the city of Monongahela in Pennsylvania. (The name "Monongalia" is either a Latinized adaptation of "Monongahela" or simply a variant spelling.)


title: Monongahela River

text: The USGS name for the river is the Monongahela River; there have been numerous alternate names, alternate spellings and misspellings in historical texts.


title: Monongahela River

text: The Monongahela is formed by the confluence of the West Fork River and its "east fork"—the Tygart Valley River—at Fairmont in north central West Virginia. From there it flows northeasterly to cross the Pennsylvania border just west of north Cheat Lake on its Cheat River tributary. Then it flows northerly across southwestern Pennsylvania, taking a bit of a detour northeast 10 miles south of Pittsburgh to approach Pittsburgh from the southeast and its confluence with the Allegheny River to form the Forks of the Ohio at "The Point" of Point State Park in Downtown Pittsburgh.


title: Monongahela River

text: Prior to early Pleistocene regional glaciation, more than 780,000 years ago, the ancestral Monongahela River (a.k.a. the Pittsburgh River) flowed northward from present-day north-central West Virginia, across western Pennsylvania and northwestern Ohio, and into the Saint Lawrence River watershed. One (or more) extensive ice sheet advance dammed the old north-flowing drainage and created a vast lake—known as Lake Monongahela—stretching from an unknown point north of present-day Beaver, Pennsylvania for ~ south to Weston, West Virginia. A river-lake with many narrow bays, its maximum water surface rose to above sea level. Over deep in places, its southwestward overflow gradually incised old drainage divides and contributed to the geological development of the present-day upper Ohio River Valley.


title: Monongahela River

text: Via the Ohio River, the river is part of the Mississippi watershed which drains to the Gulf of Mexico on the Atlantic Ocean.


title: Monongahela River

text: The river's length is 130 miles, its drainage basin is 7,340 sq.mi. and the average 30-year discharge at Elizabeth, Pennsylvania, is 9,109 cfs. It falls 3,831 ft. in elevation from its highest source to its mouth on the Ohio River. It falls 280 feet from its forks to its mouth, a stretch made navigable by locks. The mean depth is about 20 ft.


title: Monongahela River

text: In southwestern Pennsylvania, the Monongahela is met by two major tributaries: the Cheat River, which joins at Point Marion, and the Youghiogheny River, which joins at McKeesport.


title: Monongahela River

text: Major tributaries include: Becks Run, Big Sandy Creek, Buffalo Creek, Cheat River, Crooked Run, Deckers Creek, Dunkard Creek, Lick Run, Middle Fork River, Paw Paw Creek, Peters Creek, Streets Run, Turtle Creek, Tygart Valley River, West Fork River, Youghiogheny River.


title: Monongahela River

text: The river is navigable its entire length with a series of locks and dams that maintain a minimum depth of to accommodate coal-laden barges. All dams are operated by the Pittsburgh District Army Corps of Engineers. In 2006, the navigation system, operated by the U.S. Army Corps of Engineers, had nine dam-locks along of waterway. The locks overcame a change in elevation of about .


title: Monongahela River

text: According to the EPA's Toxics Release Inventory for 2010, the Monongahela ranked as the 17th most polluted river in the nation. The primary polluters were Pennsylvania iron and steel mills.


title: Monongahela River

text: The upper drainage area of the river basin is renowned for its water sports/hobbies of whitewater kayaking (and in some cases whitewater rafting) opportunities. The land here is of a very rugged plateau type which allows streams to gather sufficient water volume before they fall off the plateau and create challenging rapids. Some of the best known specific stream locations for this include:


title: Monongahela River

text: The Monongahela River valley was the site of a famous battle that was one of the first in the French and Indian War—the Braddock Expedition (May–July 1755). It resulted in a sharp defeat for two thousand British and Colonial forces against those of the French and their Native American allies.


title: Monongahela River

text: In 1817, the Pennsylvania legislature authorized the Monongahela Navigation Company to build 16 dams with bypass locks to create a river transportation system between Pittsburgh and the area that would later become West Virginia. Originally planned to run as far south as the Cheat River, the system was extended to Fairmont, and bituminous coal from West Virginia was the chief product transported downstream. After a canal tunnel through Grant's Hill in Pittsburgh was completed in 1832, boats could travel between the Monongahela River and the Western Division Canal of Pennsylvania's principal east-west canal and railroad system, the Main Line of Public Works. In 1897, the federal government took possession of the Monongahela Navigation through condemnation proceedings. Later, the dam-lock combinations were increased in size and reduced in number.


title: Monongahela River

text: Briefly linked to the Monongahela Navigation was the Youghiogheny Navigation, a slack water system of between McKeesport and West Newton. It had two dam-locks overcoming a change in elevation of about . Opening in 1850, it was destroyed by a flood in 1865.


title: Monongahela River

text: During the 19th century and well into the twentieth, the Monongahela was heavily used by industry, and several U.S. Steel plants, including the Homestead Works, site of the Homestead Strike of 1892, were built along its banks. Following the killing of several workers in the course of the strike, anarchist Emma Goldman wrote: "Words had lost their meaning in the face of the innocent blood spilled on the banks of the Monongahela." Other mills included the Edgar Thomson Works in Braddock, the first steel works in the area, the Duquesne Works and the Jones and Laughlin steel works on the South Side of Pittsburgh. Only the Edgar Thomson works remain producing steel along the river.


title: Monongahela River

text: Despite the closure of many of the mills in the 1980s and 90s, the Monogahela is still an important waterway for industry. The Mon Valley Works of U.S. Steel operates three plants, including the Edgar Thomson plant for basic steel marking, the Irvin plant for steel finishing, and the Clairton plant for coke production. Coal barges are a common sight on the river, and the railways which line either side are heavily used by freight. Other industries include power generation, chemicals, and recycling.


title: Monongahela River

text: In October 1930, severe drought caused the river flow to drop below , and in some places, it was possible to walk across the river bottom.


title: Monongahela River

text: The river was the site of a famous airplane crash that has become the subject of urban legends and conspiracy theories. Early on the morning of January 31, 1956, a B-25 bomber en route from Nellis Air Force Base in Nevada to Olmsted Air Force Base in Pennsylvania crashed into the river near the Glenwood Bridge in Homestead, Pennsylvania. The six crewmen survived the initial crash, but two of them succumbed in the cold water and drowned. Despite the relatively shallow water, the aircraft was never recovered and became known as the "ghost bomber". The "Pittsburgh Post-Gazette" published a graphical representation of the flight path and flight details in 1999. As of 2018, the bomber has not been found.


title: Buddhism and sexual orientation

text: The relationship between Buddhism and sexual orientation varies by tradition and teacher. According to some scholars, early Buddhism appears to have placed no special stigma on homosexual relations, since the subject was not mentioned.


title: Buddhism and sexual orientation

text: In the early sutras of Buddhism, "accepted or unaccepted human sexual conduct" for laypersons "is not specifically mentioned." "Sexual misconduct" is a broad term, subject to interpretation according to followers' social norms. Early Buddhism appears to have been silent concerning homosexual relations.


title: Buddhism and sexual orientation

text: According to the Pāli Canon and Āgama (the early Buddhist scriptures), it is not specified that same or opposite gender relations have inherently something to do with sexual misconduct, and some Theravada monks express that same-gender relations do not violate the rule to avoid sexual misconduct, which would mean not having sex with someone underage (thus protected by their parents or guardians), someone betrothed or married and who have taken vows of religious celibacy.


title: Buddhism and sexual orientation

text: Some later traditions feature restrictions on non-vaginal sex (some Buddhist texts mention non-vaginal sex as sexual misconduct, including men having sex with men or paṇḍaka) though some Buddhist monks interpret such texts as being about forced sex. This non-vaginal sex view is not based on what Buddha said, but from some later scriptures such as the Abhidharma texts.


title: Buddhism and sexual orientation

text: Regarding Buddhist monks, the Vinaya (code of monastic discipline) bans all sexual activity, but does so in purely physiological terms, making no moral distinctions among the many possible forms of intercourse it lists.


title: Buddhism and sexual orientation

text: Among Buddhists, there is a wide diversity of opinion about homosexuality. Buddhism teaches that sensual enjoyment and desire in general are hindrances to enlightenment, and inferior to the kinds of pleasure (see, e.g. pīti, a Pāli word often translated as "rapture") that are integral to the practice of jhāna.


title: Buddhism and sexual orientation

text: Within the earliest monastic texts such as the Vinaya ( 4th century BCE), male monks are explicitly forbidden from having sexual relations with any of the four genders: male, female, "ubhatovyañjanaka" and "paṇḍaka"; various meanings of these words are given below. Later, the Buddha allowed the ordination of women, forbade ordination to these other types of people, with exceptions to a few particular types of paṇḍaka. The Buddha's proscriptions against certain types of people joining the monastic sangha (ordained community) are often understood to reflect his concern with upholding the public image of the sangha as virtuous; in some cases, this is explicitly stated. Social acceptability was vital for the sangha, as it could not survive without material support from lay society.


title: Buddhism and sexual orientation

text: The word "ubhatovyañjanaka" is usually thought to describe people who have both male and female sexual characteristics: hermaphrodites (intersex). In the Vinaya, it is said that "ubhatovyañjanaka" should not be ordained, on account of the possibility that they would entice a fellow monk or nun into having sex. Although it has been seen by some that the category of "ubhatovyañjanaka" is of later addition to the early buddhist texts, since it does not appear in the early suttas, the "Pāṭimokkhas", nor in the early parts of the Vinaya.


title: Buddhism and sexual orientation

text: The "paṇḍaka" is a complex category that is variously defined in different Buddhist texts. In the earliest texts, the word seems to refer to a socially stigmatized class of trans-feminine and/or cross-dressing people, some of whom may have been sex workers. Paisarn Likhitpreechakul argue that these people are grouped together with groups who are excluded from ordination as well; those with physical disabilities such as deafness or dwarfism, or those who have committed crimes. "The Story of the Prohibition of the Ordination of Pandaka" from the Vinaya claims that the ban is a response to the example of a paṇḍaka monk with a desire to have sex. Being refused by other monks, he had sex with animal handlers, who then told the wider community and brought disgrace upon the sangha. Since the word "paṇḍaka" does not appear in either of the early Suttas nor in the early parts of the Vinaya, it has been seen by some as a possible indication that "pandaka's" inclusion in the Vinaya did not happen in the Buddha's lifetime but was added later .


title: Buddhism and sexual orientation

text: In the Lotus Sutra, it said Bodhisattva should not go near Paṇḍaka, as like what monk rules said in Vinaya. The Theravadin text Milinda Panha, claims that Paṇḍakas let out secrets through their imperfection.


title: Buddhism and sexual orientation

text: Some modern commentators interpret the word "ubhatovyanjañaka" as including those who are not physically intersex, but display behavioral and psychological characteristics of both sexes, such as a woman who is attracted to other women. 5th-century Buddhist writer Buddhaghosa describes "ubhatobyanjanaka" as people with the body of one sex but the "power", or gender of the other. Leonard Zwilling argues that in this account Buddhaghosa does not in fact describe "hermaphroditism" but rather bisexuality or homosexuality. Janet Gyatso pointed out that Zwilling destroys his own argument that pandakas are homosexuals when he writes, "The Vinaya, in fact, goes so far as to distinguish sexual activity between normative males from sexual relations between a socially normative male and a pandaka."


title: Buddhism and sexual orientation

text: In other texts, the term "paṇḍaka" can include those born sexually indeterminate or with no sex, eunuchs, those whose impotence changes every half month, males who gain sexual potency by absorbing other men's semen, or spying on other people having sex. It sometimes includes males or females with any sexual dysfunction, such as impotence or irregular menstrual cycles. The common element seems to be those whose sexuality is either limited physiologically, or those who are sexually impotent. Together these impotence types are almost always portrayed negatively as a pariah class, especially in the earliest texts. In modern contexts, paṇḍaka is sometimes alleged to include lesbians, gay men, and transgender and intersex people, although in ancient times, a man who sexually penetrated another man or a paṇḍaka was not himself considered a paṇḍaka.


title: Buddhism and sexual orientation

text: Some texts of the Abhidharma state that a paṇḍaka cannot achieve enlightenment in their own lifetime, but must wait for reincarnation as a man or woman. Ananda — Buddha's cousin and disciple — was said to be a paṇḍaka in one of his many previous lives, as was the Buddhist nun Isidāsī (from the Therigatha). In both cases, birth as a paṇḍaka was a result of poor karma, and the idea that being a paṇḍaka stems from bad behaviour in a previous life is common in Buddhist literature. Asanga and Vasubandhu discussed if a pandaka was able to be enlightened or not.


title: Buddhism and sexual orientation

text: In the Samantapasadika, a work of the 5th century CE Theravadin commentator and scholar Buddhaghosa's, paṇḍaka are described as being filled with defiling passions ("ussanakilesa"), unquenchable lusts ("avapasantaparilaha") and are dominated by their libido ("parilahavegabhibhuta"). The 4th century Mahayana Buddhist writers Vasubandhu and Asanga contend that the paṇḍaka has no discipline for spiritual practice, due to their defiling passions of both male and female sexes. They lack the moral fortitude to counter these passions because they lack modesty and shame. Incapable of showing restraint, such a being is abandoned by their parents and lacking such ties are unable to hold strong views. Asanga, like Vasubandhu, refuses the pandaka recognition as a layman on the grounds that such persons are unfit to associate with or serve the Sangha. Asanga, however, considers them capable to practice the path of a layman individually if they so desire, but without receiving recognition as a layman or being introduced in the sangha. A position similar to Asanga view was also featured in the Lotus Sutra, where sangha members were advised to avoid the paṇḍaka.


title: Buddhism and sexual orientation

text: Buddha's proscriptions against certain types of people joining the monastic sangha (ordained community) are often understood to reflect his concern with upholding the public image of the sangha as virtuous. Peter Jackson, the scholar of sexual politics and Buddhism in Thailand, speculates that the Buddha was initially reluctant to allow women to join the sangha for this reason. Jackson explains:


title: Buddhism and sexual orientation

text: Buddhism, the middle path, has always been concerned with the maintenance of social order and since the Buddha's time the sangha has never claimed to provide a universal vehicle for the spiritual liberation of all individuals in society, explicitly excluding those who are considered to reflect badly on the monkhood in terms of prevailing social norms and attitudes.


title: Buddhism and sexual orientation

text: Social acceptability was vital for the sangha, as it could not survive without material support from lay society.


title: Buddhism and sexual orientation

text: Several Theravada Buddhist texts state that the members of the paṇḍaka are excluded from a variety of Buddhist practices (in addition to ordination):


title: Buddhism and sexual orientation

text: Classical Mahayana scholars like Shantideva and Aśvaghoṣa considered non-vagina sex (including men sex with men) to be sexual misconduct. Shantideva based his views on quotes from the Saddharma-smrtyupasthana Sutra. According to Mizuno Kōgen's study, Saddharma-smrtyupasthana Sutra is related to Ghoṣa's Abhidharmāmṛtarasaśāstra, and is compiled by Sarvastivada sect (possibly from someone related to Ghoṣa after the 2nd century). In the Great Treatise on the Perfection of Wisdom (Sanskrit: Mahā-prajñāpāramitā-śāstra), the Madhyamaka scholar Nagarjuna mentioned the non-vagina sex restriction as based on coerced action toward one's own spouse.


title: Buddhism and sexual orientation

text: In contrast, later texts, particularly Tibetan Buddhist writings, occasionally value paṇḍaka positively for their "middleness" and balance. The paṇḍaka in these Tibetan works is translated with the term "ma ning" — "genderless" or "without genitals". The 13th-century Tibetan monk Gyalwa Yang Gönpa, who was one of the significant figures in the early Drukpa Kagyu sect, writes about "ma ning" as a balanced state between maleness and femaleness. Yang Gönpa describes "ma ning" as "the abiding breath between male exhalation and female inhalation" and "the balanced yogic channel, as opposed to the too tight male channel, and the too loose female one".


title: Buddhism and sexual orientation

text: Most Mahayana teachings assert that all beings who correctly practice the dharma may reach enlightenment, since all possess innate Buddha nature. Enlightenment being achievable even in a single life.


title: Buddhism and sexual orientation

text: Historically, Gampopa (12th century), one of the main early masters of the Kagyu school of Tibetan Buddhism, followed the Indian Buddhist tradition, starting with the 3rd-century Hinayana texts of Vasubandhu, and stated that oral and anal sex, whether with a man or a woman, are violations of the third precept regarding inappropriate sexual behavior. Longchenpa, the 13th-century founder of the Nyingma school, citing the 3rd-century Mahayana texts of the Indian master Asanga, elaborated that inappropriate sexual behavior also include the hands among inappropriate parts of the body for sexual activity. In the same way, Gelug predecessor Je Tsongkhapa also adheres to such rules in his studies.


title: Buddhism and sexual orientation

text: Various contemporary teachers of Tibetan Buddhist lineages, including the 17th Gyalwang Karmapa, have offered understanding towards LGBTQ people while noting that same-sex relationships do not necessarily constitute misconduct for lay people. The Dalai Lama has maintained the views of Je Tsongkhapa


title: Buddhism and sexual orientation

text: The 14th Dalai Lama has "voiced his support for the full recognition of human rights for all people, regardless of sexual orientation," while noting that from a Buddhist point of view, lesbian and gay sex is "generally considered sexual misconduct." In the most recent interview with the Dalai Lama on this topic (March 10, 2014), the Dalai Lama said gay marriage is OK provided it's not in contradiction with the values of one's chosen religion.


title: Buddhism and sexual orientation

text: The Dalai Lama has also stated that any sex other than penis-vagina intercourse with one's own monogamous partner, including oral sex, anal sex, and masturbation is improper from the Buddhist perspective. In his 1996 book "Beyond Dogma", he states, "A sexual act is deemed proper when the couples use the organs intended for sexual intercourse and nothing else... homosexuality, whether it is between men or between women, is not improper in itself. What is improper is the use of organs already defined as inappropriate for sexual contact."


title: Buddhism and sexual orientation

text: The Dalai Lama cited the Indian Buddhist texts of Vasubandhu, Asanga, and Ashvaghosha as his sources concerning what constitutes inappropriate sexual behavior. In 1997 he stated: "Sexual organs were created for reproduction between the male element and the female element — and everything that deviates from that is not acceptable from a Buddhist point of view." The Dalai Lama has repeatedly said to LGBT groups that he can't rewrite the texts. He thinks that this is the type of issue that would need to be discussed by a council of Buddhist elders from all Buddhist traditions.


title: Buddhism and sexual orientation

text: Peter Jackson, an Australian scholar of sexual politics and Buddhism in Thailand, writes that "Buddhism is a complex tradition and there is no single canonical or scripturally sanctioned position on homosexuality." Thailand is one of several countries with a large population of Theravada Buddhist.


title: Buddhism and sexual orientation

text: In traditional Thai Buddhist accounts of sexuality, "[sexual] actions and desires have an involuntary cause [and] do not themselves accrue any future karmic consequences. They are the outworking of past karma, not sources for the accumulation of future karma. According to Bunmi, homosexual activity and the desire to engage in homosexual activity fall into this category and are not sinful and do not accrue karmic consequences." Jackson writes that this understanding of homosexuality "prevailed in Thailand until recent decades."


title: Buddhism and sexual orientation

text: In the 1980s in Thailand, during the AIDS epidemic, there was "a shift in Buddhist attitudes from relative tolerance of homosexuality to condemnation." These views were "unprecedented in recent Thai history." During this time there were two ways Buddhists viewed homosexuality: in the sympathetic view, it was said that homosexuality arose from the karma of previous lifetimes; in the intolerant view, it was seen as arising from immoral conduct in one's present life.


title: Buddhism and sexual orientation

text: In 1989, the supreme governing body of the Thai sangha affirmed that "gays" (here translated from Thai "kathoey") are prohibited from being ordained. Their declaration has apparently gone unheeded in some quarters, as Phra Pisarn Thammapatee (AKA Phra Payom Kalayano), one of the most eminent monks in the country, demanded in 2003 that 1,000 gay monks be ousted from the sangha, and that better screening processes are put in place to keep out any gay postulants.


title: Buddhism and sexual orientation

text: Recently, Phra Payom Kalayano, an eminent monk and abbot, affirmed the rights of gay monks to join the Sangha: "In the past, katoey had no hope of being ordained because the rules were stricter and society was less open minded. But they have just as much right as anyone else to join the monkhood." This view has been affirmed by other Thai Theravada monks. But it's not known if that statement given by Phra Payom was supported only by some monks of his sangha or by official structures of his Theravada school. There is no information if any other Thai branches of Theravada has changed their attitude towards LGBT ordination because other Theravada branches does not seem to have changed their attitude on this issue.


title: Buddhism and sexual orientation

text: Several writers have noted the strong historical tradition of open bisexuality and homosexuality among male Buddhist institutions in Japan. When the Tendai priest Genshin harshly criticised homosexuality as immoral, others mistook his criticism as having been because the acolyte wasn't one's own. "Chigo Monogatari" (稚児物語), "acolyte stories" of love between monks and their "chigo" were popular, and such relationships appear to have been commonplace, alongside sex with women.


title: Buddhism and sexual orientation

text: Western Christian travellers to Japan from the 16th century have noted (with distaste) the prevalence and acceptance of forms of homosexuality among Japanese Buddhists—Jesuit priest Francis Cabral wrote in 1596 that "abominations of the flesh" and "vicious habits" were "regarded in Japan as quite honourable; men of standing entrust their sons to the bonzes to be instructed in such things, and at the same time to serve their lust".


title: Buddhism and sexual orientation

text: A 17th-century Japanese Buddhist scholar, Kitamura Kigin(北村季吟,), wrote that Buddha explained the pursuit of homosexuality over heterosexuality among priests:


title: Buddhism and sexual orientation

text: It has been the nature of men's hearts to take pleasure in a beautiful woman since the age of male and female gods, but to become intoxicated by the blossom of a handsome youth... would seem to be both wrong and unusual. Nevertheless, the Buddha preached that <nowiki>[Mount]</nowiki> Imose was a place to be avoided and the priests of the law entered this Way as an outlet for their feelings, since their hearts were, after all, made of neither stone nor wood. Like water that plunges from the peak of Tsukubane to form the deep pools of the Minano River, this love has surpassed in depth the love between women and men in these latter days. It plagues the heart not only of courtier and aristocrat but also of brave warriors. Even the mountain dwellers who cut brush for fuel have learned to take pleasure in the shade of young saplings." — Wild Azaleas (1676)


title: Buddhism and sexual orientation

text: A later Japanese legend attributed the introduction of monastic homosexuality to Japan to Shingon founder Kukai, although scholars now dismiss the veracity of this assertion, pointing out his strict adherence to the Vinaya. Nonetheless, the legend served to "affirm same-sex relation between men and boys in 17th century Japan."


title: Buddhism and sexual orientation

text: About Buddhism and homosexuality in China, scholar A. L. De Silva writes, "Generally the attitude has been one of tolerance. Matteo Ricci, the Jesuit missionary who lived in China for 27 years from 1583, expressed horror at the open and tolerant attitude that the Chinese took to homosexuality and naturally enough saw this as proof of the degeneracy of Chinese society."


title: Buddhism and sexual orientation

text: Venerable Hsing Yun, one of the premier figures in contemporary Chinese Buddhism, has stated that Buddhism should never teach intolerance toward homosexuality, and that people should expand their minds.


title: Buddhism and sexual orientation

text: Hsing Yun is a best-selling author throughout the Sinosphere, as well as an advocate of Humanistic Buddhism, an approach to reform Chinese Buddhism to match the needs of contemporary laypeople. As such, his views may not be wholly reflective of the older Buddhist views in China. However, at an earlier point (1998), he is quoted as remarking that "according to Buddhism, any emotional involvement, whether homosexual or heterosexual, is a form of attachment ... and, therefore, is a source of suffering," and when the topic of homosexuality, in particular, came up "the master spoke more strongly, calling homosexuality a perverted view ("xiejian")."


title: Buddhism and sexual orientation

text: An example of an older view in opposition to homosexuality is given by the traditional master, Hsuan Hua, an important figure for Buddhism in both China and the United States. Master Hsuan Hua stated that homosexuality "plants the seeds that lead to rebirth in the lower realms of existence". In his commentary on the Sutra of Forty-two Chapters, he described homosexuality as behavior caused by confusion, as the product of sexual desires.


title: Buddhism and sexual orientation

text: Some Mahayana Buddhist leaders were active in the movement for same-sex marriage rights in Taiwan which legalized same-sex marriages in 2019.


title: Buddhism and sexual orientation

text: In contrast to Buddhism in Asia, modern Buddhism in the Western world (European Union, United States of America, Canada, Australia and New Zealand) is typically associated with a concern for social equality—partly as a result of its largely middle-class intellectual membership base, and its philosophical roots in freethought and secular humanism. When applying Buddhist philosophy to the question of homosexuality, western Buddhists often emphasize the importance the Buddha placed on tolerance, compassion, and seeking answers within one's self. They stress these overarching values rather than examining specific passages or texts. As a result, western Buddhism is often relatively gay-friendly, especially since the 1990s. As an interpretation of what is sexual misconduct is an individual decision and not subject to judgment by any central authority, a view of accepting all peoples, but rejecting certain types of sexual acts is more predominant. LGBT people such as Issan Dorsey, Caitriona Reed, Pat Enkyo O'Hara and Soeng Hyang have been ordained as Buddhist monastics and clergy.


title: Buddhism and sexual orientation

text: The Dzogchen Ponlop Rinpoche, holder of the Karma Kagyu and Nyingma lineages, in a 2008 talk delivered to LGBT Dharma practitioners at the Shambhala Meditation Center of New York, stressed that for vajrayana lay practitioners, homosexual relationships are no better or worse than heterosexual relationships and that only unhealthy relationships, in general, are to be avoided. Both the Nalandabodhi sangha, which was founded by The Dzogchen Ponlop Rinpoche, and the Shambhala sanghas founded by Chögyam Trungpa Rinpoche have stated that they are welcoming of all sexual orientations. The Shambhala Meditation Center of New York hosts a weekly practice group, Queer Dharma, specifically catering to the needs of the LGBTQ Buddhist community. According to the Danish Karma Kagyu Lama Ole Nydahl, Buddha saw homosexuality as circumstances making life more difficult, but also explained the reason for homosexuality could be aversion against the opposite sex in a former life. Nydahl says however that sexual orientation is not really important in order to practice Buddhism. Well known Zen Buddhist, Thich Nhat Hanh, notes the spirit of Buddhism is inclusiveness and states "when you look at the ocean, you see different kinds of waves, many sizes and shapes, but all the waves have water as their foundation and substance. If you are born gay or lesbian, your ground of being in the same as mine. We are different, but we share the same ground of being."


title: Buddhism and sexual orientation

text: The U.S. branch of Soka Gakkai International, a Japan-based new religious movement (Japanese new religion) influenced by Nichiren Buddhism, announced in 1995 that they would start holding wedding ceremonies for same-sex couples, and in 2001 established a conference for LGBT members and their supporters. A Buddhist temple in Salt Lake City connected with Jodo Shinshu, another Japanese school of Buddhism, also holds religious rites for same-sex couples.


title: Buddhism and sexual orientation

text: Another Buddhist organization founded in the West, Juniper Foundation, wrote an article "A Buddhist Vote for Same Sex Marriage" demonstrating how Buddhist thinking embraces same-sex marriage:


title: Buddhism and sexual orientation

text: The heart of Buddhist thought is its insight philosophy, which uses critical inquiry to challenge dogma and to reveal how seemingly fixed ideas are more arbitrary than we might think. Applying this philosophy, we see that social customs are not fixed laws but evolving conventions that serve a purpose in a particular culture and time. Marriage is one of these conventions. It is not a rigid law but a social custom that evolves.


title: Buddhism and sexual orientation

text: In western traditions, there is widescale support for LGBT rights including the European Buddhist Union, the Buddhist Churches of America, many Shin Buddhist groups, and Zen leaders such as Thich Nhat Hanh. In a PEW research poll, 88% of American Buddhists stated that homosexuality should be accepted. This was a higher level of support than any other religious group studied. On 2012, the Australian branch of Buddhism voiced its support for same-sex marriage in a hearing of the Australian Parliament that sought to gather views on whether to legalize same-sex marriage.


title: Taegeuk

text: Taegeuk (, ) is a Korean term cognate with the Chinese term Taiji (Wade-Giles spelling: "T'ai-chi"), meaning "supreme ultimate", although it can also be translated as "great polarity / duality". The symbol was chosen for the design of the Korean national flag in the 1880s, swapping out the black and white color scheme often seen in most "taijitu" illustrations and substituting blue and red, respectively, along with a horizontal separator, as opposed to vertical.


title: Taegeuk

text: South Koreans commonly refer to their national flag as "" (Hangeul: , with "gi; 기" meaning "flag" or "banner"). This particular color-themed taegeuk symbol (i.e. using blue and red) is typically associated with Korean tradition and represents balance in the universe; the red half represents positive cosmic forces, and the blue half represents the complementary or opposing, negative cosmic forces. It is also used in Korean shamanism, Confucianism, Taoism and Buddhism.


title: Taegeuk

text: The Taegeuk diagram has been existent for the majority of written Korean history. The origins of the interlocking-sinusoid design in Korea can be traced to as early as the Goguryeo or Silla period, e.g. in the decoration of a sword, dated to the 5th or 6th century, recovered from the grave of Michu of Silla, or an artifact with the taegeuk pattern of similar age found in the Bogam-ri tombs of Baekje at Naju, South Jeolla Province in 2008. In the compound of Gameunsa, a temple built in AD 628 during the reign of King Jinpyeong of Silla, a stone object, perhaps the foundation of a pagoda, is carved with the taegeuk design.


title: Taegeuk

text: In Gojoseon, the ancient kingdom of Joseon, the design was used to express the hope for harmony of yin and yang. It is likely due to the earliest spread of ancient Chinese culture in Gojoseon, especially during the early Zhou dynasty.


title: Taegeuk

text: Today the TaeGeuk is usually associated with Korean tradition and represents balance in the universe, as mentioned in the previous section (red = 양; "yang", or positive cosmic forces, and blue = 음; "yin", or negative cosmic forces). Among its many religious connotations (Korean Confucianism; Taoism in Korea; Korean Buddhism), the taeguk is also present in Korean shamanism.


title: Taegeuk

text: The Taegeuk symbol is most prominently displayed in the center of South Korea's national flag, called the "taegeukgi", literally "taegeuk flag."(along with four of the eight trigrams used in divination). Because of the Taegeuk's association with the national flag, it is often used as a patriotic symbol, as are the colors red, blue, and black.


title: Taegeuk

text: While the use of the taeguk and the trigrams have been used since the earliest periods of Korean history, its history goes back even further in China.


title: Taegeuk

text: The Taegeuk is a Confucian icon which symbolizes cosmic balance, and represents the constant interaction between the yin and yang, also known as "eum/yang" in Korean (음양 | 陰陽). The taegeuk symbol used on the flag originated from the Chinese Confucian classic known as "The Book of Changes" (also known as "I Ching" or "Yijing"), a book developed for use in divination.


title: Taegeuk

text: The four trigrams also originated from the I Ching; each of these trigrams represent specific Confucian virtues, cosmic elements, or family roles, in addition to seasons, compass directions, etc. The 건 "geon" trigram (☰) represents the heaven (sky), summer, south, father, and justice. The 곤 "gon" trigram (☷) symbolizes the earth (ground), winter, north, mother, and vitality, the 감 "gam" trigram (☵) the moon, autumn, west, 2nd (or middle) son, and wisdom, and the 리 "ri" trigram (☲) the sun, spring, east, 2nd (or middle) daughter, and fruition. The four trigrams supposedly move in an endless cycle from "geon" to "ri" to "gon" to "gam" and back to "geon" in their pursuit of perfection.


title: Taegeuk

text: The white background symbolizes the homogeneity, integrity and purity of the Korean people. Traditionally, Koreans often wore white clothing, earning the nickname "white-clothed people" and therefore the colour white epitomizes the Korean people.


title: Taegeuk

text: A popular variant in South Korea is the tricolored taegeuk ("sam·saeg·ui tae·geuk" or "sam·tae·geuk" ), which adds a yellow lobe or "pa" ( ). The yellow portion is taken as representing humanity, in addition to the red and blue representing earth and heaven respectively. This version with more than two colours is related to the Tibetan & Korean Buddhist symbol of Gankyil.


title: Taegeuk

text: A rendition of the tricolored TaeGeuk also appeared in the official logo of the 1988 Summer Olympics accompanied by the five Olympic rings. A similar symbol is used by the Miss Asia Pacific World Beauty Pageant, which was hosted in South Korea from its inception in 2011 until 2014, and most recently hosted in the Philippines in 2019 (the 2020 event was cancelled due to COVID-19).


title: Taegeuk

text: The first designated Paralympic logo, created for the 1988 Summer Paralympics in Seoul, was based on the traditional "pa" ( ), the spiral or sinusoid components making up the Taegeuk symbol. In March 1992, the Paralympic symbol was changed to a version utilizing only three "pa". This was not fully adopted until after the 1994 Winter Paralympics in Lillehammer, Norway, since the Lillehammer Paralympic Organizing Committee had by then already started a marketing program based on the five-pa version. The three-"pa" version remained in place from the close of the Lillehammer Games through the 2004 Summer Paralympics in Athens, Greece. The current Paralympic symbol has morphed the teardrop-shaped "pa" into more of a swoosh (similar to the Nike logo), but still employs three such colour swatches, one each of red, blue, and green.


title: Chimes at Midnight

text: Falstaff (Chimes at Midnight) (Spanish: Campanadas a medianoche) is a 1966 period comedy-drama film directed by and starring Orson Welles. The Spanish-Swiss co-production was released in the United States as Chimes at Midnight and in most of Europe as Falstaff. The film's plot centres on William Shakespeare's recurring character Sir John Falstaff and the father-son relationship he has with Prince Hal, who must choose between loyalty to his father, King Henry IV, or Falstaff.


title: Chimes at Midnight

text: Welles said that the core of the film's story was "the betrayal of friendship." It stars Welles as Falstaff, Keith Baxter as Prince Hal, John Gielgud as Henry IV, Jeanne Moreau as Doll Tearsheet and Margaret Rutherford as Mistress Quickly. The script contains text from five of Shakespeare's plays; primarily "Henry IV, Part 1" and "Henry IV, Part 2", but also "Richard II", "Henry V", and "The Merry Wives of Windsor". Ralph Richardson's narration is taken from the works of chronicler Raphael Holinshed.