Information: - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - In mathematics, the phrase up to appears in discussions about the elements of a set (say "S"), and the conditions under which subsets of those elements may be considered equivalent. The statement "elements "a" and "b" of set S are equivalent up to "X"" means that "a" and "b" are equivalent if criterion "X" (such as rotation or permutation) is ignored. That is, "a" and "b" can be transformed into one another if a transform corresponding to "X" (rotation, permutation etc.) is applied. - A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories, intensional definitions (which try to give the essence of a term) and extensional definitions (which proceed by listing the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. - In mathematics, topology (from the Greek , "place", and , "study") is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness. - In topology, two continuous functions from one topological space to another are called homotopic (from Greek "homos" "same, similar" and "topos" "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. - Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. - In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant: homeomorphic topological spaces have the same fundamental group. - In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or "holes", of a topological space. - In mathematics , particularly algebraic topology , cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and point - preserving continuous maps to the category of sets and functions . They are dual to the homotopy groups , but less studied . The p - th cohomotopy set of a pointed topological space X is defined by p ( X ) = ( X , S p ) the set of pointed homotopy classes of continuous mappings from X to the p - sphere S p. For p = 1 this set has an abelian group structure , and , provided X is a CW - complex , is isomorphic to the first cohomology group H1 ( X ) , since S1 is a K ( Z , 1 ) . In fact , it is a theorem of Hopf that if X is a CW - complex of dimension at most n , then ( X , S p ) is in bijection with the p - th cohomology group H p ( X ) . The set also has a group structure if X is a suspension \ Sigma Y , such as a sphere Sq for q \ ge1 . If X is not a CW - complex , H 1 ( X ) might not be isomorphic to ( X , S 1 ) . A counterexample is given by the Warsaw circle , whose first cohomology group vanishes , but admits a map to S1 which is not homotopic to a constant map - Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature (as number), while others are functioning as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little. A small quantity is sometimes referred to as a quantulum. - Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. - In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term "abstract algebra" was coined in the early 20th century to distinguish this area of study from the other parts of algebra. - In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spacesthat is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word "homeomorphism" comes from the Greek words "" ("homoios") = similar and "" ("morph") = shape, form. After reading the paragraphs above, we are interested in knowing the entity with which 'cohomotopy group' exhibits the relationship of 'subclass of'. Find the answer from the choices below. Choices: - algebra - area - branch - category - collection - definition - direction - division - entity - essence - field - greek - group - homeomorphism - information - invariant - isomorphism - learning - magnitude - mathematics - number - phrase - property - quantity - range - relation - rotation - shape - space - statement - stretching - structure - study - term - topology
Answer:
invariant