hcju/proofwikips huggingface dataset:
Definition:Final Topology/Definition 1
The '''final topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$''' is defined as: :$\tau = \set{U \subseteq X: \forall i \in I: \map {f_i^{-1}} U \in \tau_i} \subseteq \powerset X$
Definition:Final Topology/Definition 2
Let $\tau$ be the finest topology on $X$ such that each $f_i: Y_i \to X$ is $\tuple{\tau_i, \tau}$-continuous. Then $\tau$ is known as the '''final topology on $X$ with respect to $\family{f_i}_{i \mathop \in I}$'''.
Definition:Subfamily
Let $I$ and $S$ be sets. Let $x: I \to S$ be an indexing function for $S$. Let $\family {x_i}_{i \mathop \in I}$ be a family indexed by $I$. Let $J \subseteq I$ be a subset of $I$. The family $\family {x_j}_{j \mathop \in J}$ indexed by $J$ is known as a '''subfamily''' of $\family {x_i}_{i \mathop \in I}$.
Definition:Accumulation Point
Let $\paren { S, \tau }$ be a topological space. Let $A \subseteq S$. === Accumulation Point of Sequence === {{:Definition:Accumulation Point/Sequence}} === Accumulation Point of Set === {{:Definition:Accumulation Point/Set}}
Definition:Strictly Smaller Set
Let $S$ and $T$ be sets. $S$ is defined as being '''strictly smaller than''' $T$ : :$(1): \quad$ There exists a bijection from $S$ to a subset of $T$ :$(2): \quad$ There exists no such bijection from $T$ to a subset of $S$. '''$S$ is strictly smaller than $T$''' can be denoted: :$S < T$
Definition:Smaller Set
Let $S$ and $T$ be sets. $S$ is defined as being '''smaller than''' $T$  there exists a bijection from $S$ to a subset of $T$. '''$S$ is smaller than $T$''' can be denoted: :$S \le T$
Definition:Strictly Larger Set
Let $S$ and $T$ be sets. $S$ is defined as being '''strictly larger than''' $T$ : :$(1): \quad$ There exists a bijection from $T$ to a subset of $S$ :$(2): \quad$ There exists no such bijection from $S$ to a subset of $T$. '''$S$ is strictly larger than $T$''' can be denoted: :$S > T$
Definition:Larger Set
Let $S$ and $T$ be sets. $S$ is defined as being ''' larger than''' $T$  there exists a bijection from $T$ to a subset of $S$. '''$S$ is larger than $T$''' can be denoted: :$S \ge T$
Definition:Comparable Sets/Cardinality
Let $S$ and $T$ be sets. Then $S$ and $T$ are '''comparable (in size)'''  either: :$S$ can be put into one-to-one correspondence with a subset of $T$ or: :$T$ can be put into one-to-one correspondence with a subset of $S$ or both. That is, if either $S$ is smaller than $T$ or $T$ is smaller than $S$.
Definition:First Order Logic with Identity
A system of '''first order logic with identity''' is a system of predicate logic with the following elements in its alphabet: ==== Connectives ==== {{begin-axiom}} {{axiom | ml= \land          | mo= :         | t = the conjunction sign }} {{axiom | ml= \lor         | mo= :         | t = the disjunction sign }} {{axiom | ml= \implies         | mo= :         | t = the conditional sign }} {{axiom | ml= \iff         | mo= :         | t = the biconditional sign }} {{axiom | ml= \neg         | mo= :         | t = the negation sign }} {{end-axiom}} ==== Quantifiers ==== {{begin-axiom}} {{axiom | ml= \exists         | mo= :         | t = the existential quantifier sign }} {{axiom | ml= \forall         | mo= :         | t = the universal quantifier sign }} {{end-axiom}} ==== Identity ==== {{begin-axiom}} {{axiom | ml= =         | mo= :         | t = the equality sign }} {{end-axiom}}
Definition:Frege Set Theory
The '''Frege system of set theory''' is a system of axiomatic set theory which has as its sole axiom the comprehension principle: {{:Axiom:Comprehension Principle}} In support of this, the various logical axioms supporting predicate logic also hold.
Definition:Zermelo Set Theory
'''Zermelo set theory''' is a system of axiomatic set theory. Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the following axioms: === The Axiom of Extension === {{:Axiom:Axiom of Extension/Set Theory}} === The Axiom of the Empty Set === {{:Axiom:Axiom of Empty Set/Set Theory}} === The Axiom of Pairing === {{:Axiom:Axiom of Pairing/Set Theory}} === The Axiom of Specification === {{:Axiom:Axiom of Specification/Set Theory}} === The Axiom of Unions === {{:Axiom:Axiom of Unions/Set Theory}} === The Axiom of Powers === {{:Axiom:Axiom of Powers/Set Theory}} === The Axiom of Infinity === {{:Axiom:Axiom of Infinity}}
Definition:Zermelo-Fraenkel-Skolem Set Theory
'''Zermelo-Fraenkel-Skolem set theory''' is a system of axiomatic set theory. It consists of Zermelo-Fraenkel set theory with the inclusion of a system of first order formulas to specify the existence of properties of sets. {{NamedforDef|Ernst Friedrich Ferdinand Zermelo|name2 = Abraham Halevi Fraenkel|name3 = Thoralf Albert Skolem|cat = Zermelo|cat2 = Fraenkel|cat3 = Skolem}}
Definition:Morse-Kelley Set Theory
'''Morse-Kelley set theory''' is a system of axiomatic set theory. It is a stronger form of Zermelo-Fraenkel-Skolem set theory which allows not only for first order formulas to specify the existence of properties of sets, but also defines properties by quantifying over properties as well as over sets.  {{NamedforDef|Anthony Perry Morse|name2 = John Leroy Kelley|cat = Morse|cat2 = Kelley}}
Definition:Von Neumann-Bernays-Gödel Set Theory
'''Von Neumann-Bernays-Gödel set theory''' is a system of axiomatic set theory. Its main feature is that it classifies collections of objects into: :sets, whose construction is strictly controlled and: :classes, which have fewer restrictions on how they may be generated. All sets are classes, but not all classes are sets.
Definition:Model of Class-Set Theory
Let $V$ be a collection of objects. Then $V$ is a '''model of class-set theory'''  it satisfies the axioms of Von Neumann-Bernays-Gödel set theory. Thus:  :The elements of $V$ are then known as sets :The subcollections of $V$ are known as classes.
Definition:Collection
A '''collection''' is an aggregation of objects whose nature is arbitrary. The precise nature and structure of the '''collection''' itself is left unspecified. Hence the word '''collection''' can be conveniently used for an aggregation which may be either a '''set''' or a '''class''', and such that whatever is being said about it applies to both.
Definition:Element/Notation
The symbol universally used in modern mainstream mathematics to mean '''$x$ is an element of $S$''' is: :$x \in S$ Similarly, $x \notin S$ means '''$x$ is ''not'' an element of $S$'''. The symbol can be reversed: :$S \ni x$ means '''$S$ has $x$ as an element''', that is, $x$ is an element of $S$ but this is rarely seen. Some texts (usually older ones) use $x \mathop {\overline \in} S$ or $x \mathop {\in'} S$ instead of $x \notin S$.
Definition:Element/Also known as
The term '''member''' is sometimes used as a synonym for element (probably more for the sake of linguistic variation than anything else). In the contexts of geometry and topology, '''elements''' of a set are often called '''points''', in particular when they ''are'' (geometric) points. $x \in S$ can also be read as: * '''$x$ is in $S$''' * '''$x$ belongs to $S$''' * '''$S$ includes $x$''' * '''$x$ is included in $S$''' * '''$S$ contains $x$''' However, '''beware''' of this latter usage: '''$S$ contains $x$''' can also be interpreted as '''$x$ is a subset of $S$'''. Such is the scope for misinterpretation that it is '''mandatory''' that further explanation is added to make it clear whether you mean subset or element.
Definition:Element/Class
Let $S$ be a class. An '''element of $S$''' is a member of $S$.
Definition:Class Equality/Definition 1
$A$ and $B$ are '''equal''', denoted $A = B$, : :$\forall x: \paren {x \in A \iff x \in B}$ where $\in$ denotes class membership.
Definition:Class Equality/Definition 2
$A$ and $B$ are '''equal''', denoted $A = B$, : :$A \subseteq B$ and $B \subseteq A$ where $\subseteq$ denotes the subclass relation.
Definition:First-Order Property of Sets
A '''first-order property of sets''' is a property defined by a logical formula whose domain is entirely over sets. Thus the quantifiers $\forall x$ and $\exists x$ are valid when $x$ is a set, but $\forall A$ and $\exists A$ are ''not'' valid when $A$ is a class. Hence we allow, for example: :$\map \phi {A_1, A_2, \ldots, A_n, x}$ to be a logical formula whose variables $A_1, A_2, \ldots, A_n$ represent classes, and are all free, and $x$ is the only free variable representing a set.
Definition:Swelled Class
Let $A$ denote a class. Then $A$ is a '''swelled class'''  every subclass of every element of $A$ is also an element of $A$.
Definition:Supercomplete Class
Let $A$ denote a class. Then $A$ is a '''supercomplete class'''  both: :$A$ is transitive and: :$A$ is swelled.
Definition:Von Neumann-Bernays-Gödel Axioms
=== The Axiom of Extension === {{:Axiom:Axiom of Extension/Class Theory}} === The Axiom of Specification === {{:Axiom:Axiom of Specification/Class Theory}}
Definition:Basic Universe
A '''basic universe''' $V$ is a universal class which satisfies the following axioms: {{:Definition:Basic Universe Axioms}}
Definition:Basic Universe Axioms
=== $\text A 1$: Axiom of Transitivity === {{:Axiom:Axiom of Transitivity}} === $\text A 2$: Axiom of Swelledness === {{:Axiom:Axiom of Swelledness}} === $\text A 3$: Axiom of the Empty Set === {{:Axiom:Axiom of Empty Set/Class Theory}} === $\text A 4$: Axiom of Pairing === {{:Axiom:Axiom of Pairing/Class Theory}} === $\text A 5$: Axiom of Unions === {{:Axiom:Axiom of Unions/Class Theory}} === $\text A 6$: Axiom of Powers === {{:Axiom:Axiom of Powers/Class Theory}}
Definition:Zermelo-Fraenkel Universe
A '''Zermelo-Fraenkel universe''' is a basic universe which also satisfies the axiom of infinity and the axiom of substitution: {{:Definition:Basic Universe Axioms}} === $\text A 7$: Axiom of Infinity === {{:Axiom:Axiom of Infinity/Class Theory}} === $\text A 8$: Axiom of Substitution === {{:Axiom:Axiom of Substitution}}
Definition:Zermelo Universe
A '''Zermelo universe''' is a basic universe which also satisfies the axiom of infinity: {{:Definition:Basic Universe Axioms}} === $\text A 7$: Axiom of Infinity === {{:Axiom:Axiom of Infinity/Class Theory}}
Definition:Empty Class (Class Theory)
A class is defined as being '''empty'''  it has no elements. That is: :$\forall x: x \notin A$ or: :$\neg \exists x: x \in A$ The '''empty class''' is usually denoted $\O$ or $\emptyset$.
Definition:Singleton Class
Let $a$ be a set. The class $\set a$ is a '''singleton (class)'''. It can be referred to as '''singleton $a$'''.
Definition:Ordered Pair/Wiener Formalization
The concept of an ordered pair can be formalized by the definition: :$\tuple {a, b} := \set {\set {\O, \set a}, \set {\set b} }$
Definition:Ordered Pair/Empty Set Formalization
The concept of an ordered pair can be formalized by the definition: :$\tuple {a, b} := \set {\set {\O, a}, \set {\set \O, b} }$
Definition:Set Intersection/Class
Let $A$ be a class. The '''intersection of $A$''' is: :$\displaystyle \bigcap A := \set {x: \forall y \in A: x \in y}$ That is, the class of all elements which belong to all the elements of $A$.
Definition:Class Union
Let $A$ and $B$ be two classes. The '''(class) union''' $A \cup B$ of $A$ and $B$ is defined as the class of all sets $x$ such that either $x \in A$ or $x \in B$ or both: :$x \in A \cup B \iff x \in A \lor x \in A$ or: :$A \cup B = \set {x: x \in A \lor x \in B}$
Definition:Class Intersection
Let $A$ and $B$ be two classes. The '''(class) intersection''' $A \cap B$ of $A$ and $B$ is defined as the class of all sets $x$ such that $x \in A$ and $x \in B$: :$x \in A \cap B \iff x \in A \land x \in A$ or: :$A \cap B = \set {x: x \in A \land x \in B}$
Definition:Power Set/Class Theory
The '''power set''' of a set $S$ is the class of all the subsets of $S$: :$\powerset S := \set {T: T \subseteq S}$
Definition:Polygon/Adjacent/Side to Vertex
Each vertex of a polygon is formed by the intersection of two sides. The two sides that form a particular vertex are referred to as the '''adjacents''' of that vertex, or described as '''adjacent to''' that vertex.
Definition:Polygon/Adjacent/Vertex to Side
Each side of a polygon intersects two other sides, and so is terminated at either endpoint by two vertices. The two vertices that terminate a particular side are referred to as the '''adjacents''' of that side, or described as '''adjacent to''' that side.
Definition:Polygon/Adjacent/Vertices
Those two vertices are described as '''adjacent to each other'''.
Definition:Polygon/Adjacent/Sides
Two sides of a polygon that meet at the same vertex are '''adjacent''' to each other.
Definition:Diophantine Analysis
'''Diophantine analysis''' is the branch of number theory which is concerned with the solution of Diophantine equations.
Definition:Generatrix/Also known as
A '''generatrix''' of a surface can also be referred to as a '''generator''' or '''generator element'''.
Definition:Vector Product
=== Dot Product === {{:Definition:Dot Product}} === Cross Product === {{:Definition:Vector Cross Product}} === Outer Product === {{:Definition:Outer Product}}
Definition:Stochastic Calculus
'''Stochastic calculus''' is a generalization of calculus to the subject of stochastic processes.
Definition:QEF
The initials of '''Quod Erat Faciendum''', which is {{WP|Latin|Latin}} for '''which was to be done'''. These initials were traditionally added to the end of a geometric construction.
Definition:QEI
The initials of '''Quod Erat Inveniendum''', which is {{WP|Latin|Latin}} for '''which was to be found'''. These initials were traditionally added after the completion of a calculation (either arithmetic or algebraic).
Definition:Quintal Notation
'''Quintal notation''' is the technique of expressing numbers in base $5$. That is, every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 5^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4}$
Definition:Random
=== Random Error === {{:Definition:Random Error}} === Random Number === {{:Definition:Random Number}} === Random Sample === {{:Definition:Random Sample}} === Random Selection === {{:Definition:Random Selection}} === Random Variable === {{:Definition:Random Variable}} === Random Vector === {{:Definition:Random Vector}} === Random Walk === {{:Definition:Random Walk}}
Definition:Decade
A '''decade''' is a set or ordered tuple of $10$ numbers. Its most common usage is as a period of time: === Decade (Time Period) === {{:Definition:Decade (Time)}}
Definition:Parastrophe
=== $(1-3)$ Parastrophe === {{:Definition:(1-3) Parastrophe}} === $(2-3)$ Parastrophe === {{:Definition:(2-3) Parastrophe}} === $(1-2)$ Parastrophe === {{:Definition:(1-2) Parastrophe}}
Definition:Continuous Mapping (Normed Vector Space)
Let $M_1 = \struct{X_1, \norm {\,\cdot\,}_{X_1} }$ and $M_2 = \struct{X_2, \norm {\,\cdot\,}_{X_2} }$ be normed vector spaces. Let $f: X_1 \to X_2$ be a mapping from $X_1$ to $X_2$. Let $a \in X_1$ be a point in $X_1$. === Continuous at a Point === {{Definition:Continuous Mapping (Normed Vector Space)/Point}} === Continuous on a Space === {{Definition:Continuous Mapping (Normed Vector Space)/Space}}
Definition:Continuous Mapping (Normed Vector Space)/Point
{{Definition:Continuous Mapping (Normed Vector Space)/Point/Definition 1}}
Definition:Continuous Mapping (Normed Vector Space)/Point/Definition 1
'''$f$ is continuous at (the point) $a$ (with respect to the norms $\norm {\,\cdot\,}_{X_1}$ and $\norm {\,\cdot\,}_{X_2}$)''' : :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in X_1: \norm {x - a}_{X_1} < \delta \implies \norm {\map f x - \map f a}_{X_2} < \epsilon$ where $\R_{>0}$ denotes the set of all strictly positive real numbers.
Definition:Continuous Mapping (Normed Vector Space)/Space
{{Definition:Continuous Mapping (Normed Vector Space)/Space/Definition 1}}
Definition:Continuous Mapping (Normed Vector Space)/Space/Definition 1
$f$ is '''continuous from $\struct{X_1, \norm {\,\cdot\,}_{X_1} }$ to $\struct{X_2, \norm {\,\cdot\,}_{X_2} }$'''  it is continuous at every point $x \in X_1$.
Definition:Statics
'''Statics''' is the branch of mechanics investigating the behaviour of forces of bodies at rest.
Definition:Limit of Function (Normed Vector Space)/Epsilon-Delta Condition
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < \norm {x - c}_{X_1} < \delta \implies \norm {\map f x - L}_{X_2} < \epsilon$ That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that ''every'' point in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of some point $L$ in the codomain of $f$.
Definition:Limit of Function (Normed Vector Space)/Epsilon-Ball Condition
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \map f {B_\delta \left({c; \norm {\,\cdot\,}_{X_1}}\right) \setminus \set c} \subseteq \map {B_\epsilon} {L; \norm {\,\cdot\,}_{X_2}}$. where: : $\map {B_\delta} {c; \norm {\,\cdot\,}_{X_1}} \setminus \set c$ is the deleted $\delta $-neighborhood of $c$ in $M_1$ : $B_\epsilon \left({L; \norm {\,\cdot\,}_{X_2}}\right)$ is the open $\epsilon$-ball of $L$ in $M_2$. That is, for every open $\epsilon$-ball of $L$ in $M_2$, there exists a deleted $\delta $-neighborhood of $c$ in $M_1$ whose image is a subset of that open $\epsilon$-ball.
Definition:Body/Rigid
A '''rigid body''' is a body $B$ with the property that the distance between any two points of $B$ remains constant regardless of any external forces.
Definition:Motion
A body is defined as being '''in motion''' if it is not stationary with respect to the given frame of reference. Thus '''motion''' is the state of '''changing position'''.
Definition:Ceteris Paribus
'''Ceteris paribus''' is a Latin phrase meaning '''all other things being equal'''. It is also translated as '''in the absence of any other change''' (in the conditions of a theorem, for example). It is often used in logical arguments in natural language when expressing a general comparison between two effects, and is a plea to the hearer to suspend counterarguments based on differences in the general situations.
Definition:Natural Place
According to the ancient Greek philosophers, everything had a position in the universe where it was most comfortable. That position was referred to as its '''natural place'''. Hence, when displaced from its '''natural place''', it would eagerly try to return to that '''natural place'''. Hence: :a stone, being made of earth, when released from a hand holding it in midair, will push downwards in order to return as soon as possible to the centre of the universe, that is, Earth. :tongues of fire will shoot upwards, so as to rise to its '''natural place''' above the sphere of air :bubbles of air will rise up through water to its '''natural place''' above the sphere of water.
Definition:Vacuum
A '''vacuum''' is the physical state of being completely empty of matter. It is a physical ideal, and has never been (and probably can never be) fully realised.
Definition:Inclined Plane
An '''inclined plane''' is a plane surface which is neither horizontal nor vertical.
Definition:Gravity
'''Gravity''' is the tendency of bodies with mass to attract each other.
Definition:Gravity/Gravitational Force
The '''gravitational force''' on a body $B$ is the force which is exerted on $B$ as a result of the gravitational field whose influence it is under.
Definition:Limit Point/Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space. Let $Y \subseteq X$ be a subset of $X$. === Limit Point of Set === {{:Definition:Limit Point/Normed Vector Space/Set}} === Limit Point of Sequence === {{:Definition:Limit Point/Normed Vector Space/Sequence}}
Definition:Displacement/Dimension
The dimension of '''displacement''' is length $\mathsf L$.
Definition:Mass/Units
The units of measurement of '''mass''' are as follows: * The SI unit of '''mass''' is the kilogram $\mathrm {kg}$. * The CGS unit of '''mass''' is the gram $\mathrm g$. * The FPS unit of '''mass''' is the pound $\mathrm {lb}$. Thus: :$1 \ \mathrm {kg} = 10^3 \ \mathrm g$ :$1 \ \mathrm {lb} = 0.453 \, 592 \, 37 \ \mathrm{kg}$
Definition:Gram/Symbol
The symbol for the '''gram''' is $\mathrm g$.
Definition:CGS/Base Units
{| class="wikitable" style="margin:1em auto 1em auto" |+ CGS base units |- !Name ! Unit symbol ! Dimension ! Symbol |- ! centimetre | $\mathrm{cm}$ | Length | $l$ |- ! gram | $\mathrm g$ | Mass | $m$ |- ! second | $\mathrm s$ | Time | $t$ |- |}
Definition:CGS/Derived Units/Dyne
The '''dyne''' is the '''CGS''' unit of force: :$1 \ \mathrm {dyn} = 1 \ \mathrm g \ \mathrm{cm} \ \mathrm s^{-2}$
Definition:MKS/Derived Units
=== Newton === {{:Definition:Newton (Unit)}}
Definition:Expectation/Discrete
Let $X$ be a discrete random variable. The '''expectation of $X$''' is written $\expect X$, and is defined as: :$\expect X := \displaystyle \sum_{x \mathop \in \image X} x \, \map \Pr {X = x}$ whenever the sum is absolutely convergent, that is, when: :$\displaystyle \sum_{x \mathop \in \image X} \size {x \, \map \Pr {X = x} } < \infty$
Definition:Limit Point/Normed Vector Space/Sequence
Let $L \in Y$. Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $Y \setminus \set L$. Let $\sequence {x_n}_{n \mathop \in \N}$ converge to $L$.  Then $L$ is a '''limit of  $\sequence {x_n}_{n \mathop \in \N}$ as $n$ tends to infinity''' which is usually written: :$\displaystyle L = \lim_{n \mathop \to \infty} x_n$
Definition:Artificial Intelligence
'''Artificial intelligence''' is a branch of computer science which studies intelligent behavior, and attempts to model it via computing techniques.
Definition:Deduction
'''Deduction''' is the process in logic which derives new true statements from a set of premises, themselves assumed true.
Definition:Temporal Logic
'''Temporal logic''' is a subcategory of modal logic which introduces time, by defining the concepts: :sometimes :always.
Definition:Limit Point/Normed Vector Space/Set
Let $\alpha \in X$.  Then $\alpha$ is a '''limit point of $Y$'''  ''every'' deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $Y$: :$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} \alpha \setminus \set \alpha \cap Y \ne \O$ that is: :$\forall \epsilon \in \R_{>0}: \set {x \in Y: 0 < \norm {x - \alpha} < \epsilon} \ne \O$ Note that $\alpha$ does not have to be an element of $A$ to be a '''limit point'''.
Definition:Abel Summation Method
{{Help|It is difficult finding a concise and complete definition of exactly what the Abel Summation Method actually is. All and any advice as to how to implement this adequately is requested of anyone. This is what is said in the Spring encyclopedia on the page "Abel summation method":}} The series: :$\displaystyle \sum a_n$ can be summed by the Abel method ($A$-method) to the number $S$ if, for any real $x$ such that $0 < x < 1$, the series: :$\displaystyle \sum_{k \mathop = 0}^\infty a_k x^k$ is convergent and: :$\displaystyle \lim_{x \mathop \to 1^-} \sum_{k \mathop =0}^\infty a_k x^k = S$ {{help|This is what we have on Wikipedia page {{WP|Divergent_series|Divergent series}}: }} :$\displaystyle \map f x = \sum_{n \mathop = 0}^\infty a_n e^{-n x} = \sum_{n \mathop = 0}^\infty a_n z^n$ where $z = \map \exp {−x}$. Then the limit of $\map f x$ as $x$ approaches $0$ through positive reals is the limit of the power series for $\map f z$ as $z$ approaches $1$ from below through positive reals. The '''Abel sum''' $\map A s$ is defined as: :$\displaystyle \map A s = \lim_{z \mathop \to 1^-} \sum_{n \mathop = 0}^\infty a_n z^n$ {{NamedforDef|Niels Henrik Abel|cat = Abel}}
Definition:Above
In the context of real numbers, '''above''' means '''greater than'''.
Definition:Maximum Value of Real Function
=== Absolute Maximum === {{:Definition:Maximum Value of Real Function/Absolute}} === Local Maximum === {{:Definition:Maximum Value of Real Function/Local}}
Definition:Minimum Value of Real Function
=== Absolute Minimum === {{:Definition:Minimum Value of Real Function/Absolute}} === Local Minimum === {{:Definition:Minimum Value of Real Function/Local}}
Definition:Topology Generated by Synthetic Basis/Definition 3
The '''topology on $S$ generated by $\BB$''' is defined as: :$\tau = \set {U \subseteq S: \forall x \in U: \exists B \in \BB: x \in B \subseteq U}$