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Adjunction Space
In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be topological spaces, and let A be a subspace of Y. Let f : A \rightarrow X be a continuous map (called the attaching map). One forms the adjunction space X \cup_f Y (sometimes also written as X +_f Y) by taking the disjoint union of X and Y and identifying a with f(a) for all a in A. Formally, :X\cup_f Y = (X\sqcup Y) / \sim where the equivalence relation \sim is generated by a\sim f(a) for all a in A, and the quotient is given the quotient topology. As a set, X \cup_f Y consists of the disjoint union of X and ( Y-A). The topology, however, is specified by the quotient construction. Intuitively, one may think of Y as being glued onto X via the map f. Examples *A common example of an adjunction space is given when ''Y'' is a closed ''n''-ball (or ''cell'') and ''A'' is the boundary of the ball ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is Closure (mathematics), closed under the limit of a sequence, limit operation. This should not be confused with closed manifold. Sets that are both open and closed and are called clopen sets. Definition Given a topological space (X, \tau), the following statements are equivalent: # a set A \subseteq X is in X. # A^c = X \setminus A is an open subset of (X, \tau); that is, A^ \in \tau. # A is equal to its Closure (topology), closure in X. # A contains all of its limit points. # A contains all of its Boundary (topology), boundary points. An alternative characterization (mathematics), characterization of closed sets is available via sequences and Net (mathematics), net ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald Brown (mathematician)
Ronald Brown Learned Society of Wales, FLSW (born 4 January 1935 – 6 December 2024) was an English mathematician. He was a Professor in the School of Computer Science at Bangor University. He has authored many books and more than 160 journal articles. Brown died on 6 December 2024, at the age of 89. Education and career Born on 4 January 1935 in London, Brown attended Oxford University, obtaining a Bachelor of Arts, B.A. in 1956 and a D.Phil. in 1962. Brown began his teaching career during his doctorate work, serving as an assistant lecturer at the University of Liverpool before assuming the position of Lecturer. In 1964, he took a position at the University of Hull, serving first as a Senior Lecturer and then as a Reader before becoming a Professor of pure mathematics at Bangor University, then a part of the University of Wales, in 1970. Brown served as Professor of Pure Mathematics for 30 years; he also served during the 1983–84 term as a Professor for one month at Louis P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mapping Cylinder
In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function f between topological spaces X and Y is the quotient :M_f = (( ,1times X) \amalg Y)\,/\,\sim where the \amalg denotes the disjoint union, and ~ is the equivalence relation generated by :(0,x)\sim f(x)\quad\textx\in X. That is, the mapping cylinder M_f is obtained by gluing one end of X\times ,1/math> to Y via the map f. Notice that the "top" of the cylinder \\times X is homeomorphic to X, while the "bottom" is the space f(X)\subset Y. It is common to write Mf for M_f, and to use the notation \sqcup_f or \cup_f for the mapping cylinder construction. That is, one writes :Mf = ( ,1times X) \cup_f Y with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct the mapping cone Cf, obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations. Basic properties The bottom ''Y'' is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Space (topology)
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space. Intuitively speaking, the points of each equivalence class are or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space. Definition Let X be a topological space, and let \sim be an equivalence relation on X. The quotient set Y = X/ is the set of equivalence classes of elements of X. The e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inclusion Map
In mathematics, if A is a subset of B, then the inclusion map is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota(x)=x. An inclusion map may also be referred to as an inclusion function, an insertion, or a canonical injection. A "hooked arrow" () is sometimes used in place of the function arrow above to denote an inclusion map; thus: \iota: A\hookrightarrow B. (However, some authors use this hooked arrow for any embedding.) This and other analogous injective functions from substructures are sometimes called natural injections. Given any morphism f between objects X and Y, if there is an inclusion map \iota : A \to X into the domain X, then one can form the restriction f\circ \iota of f. In many instances, one can also construct a canonical inclusion into the codomain R \to Y known as the range of f. Applications of inclusion maps Inclusion maps tend to be homomorphisms of algebraic structures; thus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Diagram
350px, The commutative diagram used in the proof of the five lemma In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra. Description A commutative diagram often consists of three parts: * objects (also known as ''vertices'') * morphisms (also known as ''arrows'' or ''edges'') * paths or composites Arrow symbols In algebra texts, the type of morphism can be denoted with different arrow usages: * A monomorphism may be labeled with a \hookrightarrow or a \rightarrowtail. * An epimorphism may be labeled with a \twoheadrightarrow. * An isomorphism may be labeled with a \overset. * The dashed arrow typically represents the claim that the indicated morphism exists (whenever the rest of the diagram holds); the arrow may be optionally labeled as \e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field (mathematics), field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of category (mathematics), categories and functors by means of a universal morphism (see , below). Universal morphisms can also be thought more abstractly as Initia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Topological Spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds, with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces. As a concrete category Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor to the category of sets which assigns to each topological space the underlyin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pushout (category Theory)
In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms ''f'' : ''Z'' → ''X'' and ''g'' : ''Z'' → ''Y'' with a common domain. The pushout consists of an object ''P'' along with two morphisms ''X'' → ''P'' and ''Y'' → ''P'' that complete a commutative square with the two given morphisms ''f'' and ''g''. In fact, the defining universal property of the pushout (given below) essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are P = X \sqcup_Z Y and P = X +_Z Y. The pushout is the categorical dual of the pullback. Universal property Explicitly, the pushout of the morphisms ''f'' and ''g'' consists of an object ''P'' and two morphisms ''i''1 : ''X'' → ''P'' and ''i''2 : ''Y'' → ''P'' such that the diagram : commutes and such th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some Injective function, injective and structure-preserving map f:X\rightarrow Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map f:X\rightarrow Y is an embedding is often indicated by the use of a "hooked arrow" (); thus: f : X \hookrightarrow Y. (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
