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Category Of Compactly Generated Weak Hausdorff Spaces
In mathematics, the category of compactly generated weak Hausdorff spaces, CGWH, is a category used in algebraic topology as an alternative to the category of topological spaces, Top, as the latter lacks some properties that are common in practice and often convenient to use in proofs. There is also such a category for the CGWH analog of pointed topological spaces, defined by requiring maps to preserve base points. The articles compactly generated space and weak Hausdorff space define the respective topological properties. For the historical motivation behind these conditions on spaces, see Compactly generated space#Motivation. This article focuses on the properties of the category. Properties CGWH has the following properties: *It is complete and cocomplete. *The forgetful functor to the sets preserves small limits. *It contains all the locally compact Hausdorff spaces and all the CW complexes. *An internal Hom exists for any pairs of spaces ''X'' and ''Y''; it is denoted by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactly Generated Space
In topology, a topological space X is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other. Also some authors include some separation axiom (like Hausdorff space or weak Hausdorff space) in the definition of one or both terms, and others do not. In the simplest definition, a ''compactly generated space'' is a space that is coherent with the family of its compact subspaces, meaning that for every set A \subseteq X, A is open in X if and only if A \cap K is open in K for every compact subspace K \subseteq X. Other definitions use a family of continuous maps from compact spaces to X and declare X to be compactly generated if its topology coincides with the final topology with respect to this family of maps. And other variations of the definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Hausdorff Space
In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every Continuous function (topology), continuous map from a Compact space, compact Hausdorff space into the space is closed set, closed. In particular, every Hausdorff space is weak Hausdorff. As a separation axiom, separation property, it is stronger than T1, which is equivalent to the statement that points are closed. Specifically, every weak Hausdorff space is a T1 space, T1 space. The notion was introduced by M. C. McCord to remedy an inconvenience of working with the Category (mathematics), category of Hausdorff spaces. It is often used in tandem with compactly generated spaces in algebraic topology. For that, see the category of compactly generated weak Hausdorff spaces. k-Hausdorff spaces A is a topological space which satisfies any of the following equivalent conditions: # Each compact subspace is Hausdorff space, Hausdorff. # The diagonal \ is k-closed set, k-cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups 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 ... [...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] |
Pointed Space
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains unchanged during subsequent discussion, and is kept track of during all operations. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map f between a pointed space X with basepoint x_0 and a pointed space Y with basepoint y_0 is a based map if it is continuous with respect to the topologies of X and Y and if f\left(x_0\right) = y_0. This is usually denoted :f : \left(X, x_0\right) \to \left(Y, y_0\right). Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint. The pointed set concept is less important; it is anyway the case of a pointed discrete space. Pointed spaces are often taken a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactly Generated Space
In topology, a topological space X is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other. Also some authors include some separation axiom (like Hausdorff space or weak Hausdorff space) in the definition of one or both terms, and others do not. In the simplest definition, a ''compactly generated space'' is a space that is coherent with the family of its compact subspaces, meaning that for every set A \subseteq X, A is open in X if and only if A \cap K is open in K for every compact subspace K \subseteq X. Other definitions use a family of continuous maps from compact spaces to X and declare X to be compactly generated if its topology coincides with the final topology with respect to this family of maps. And other variations of the definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Category
In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocomplete. The existence of ''all'' limits (even when ''J'' is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other. A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits exists (i.e. limits of diagrams indexed by a finite category ''J''). Dually, a category is finitely cocomplete if all finite colimits exist. Theorems It follows from the existence theorem for limits that a category is complete if and only if it has equalizers (o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forgetful Functor
In mathematics, more specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all of the input's structure or properties mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case. Overview As an example, there are several forgetful functors from the category of commutative rings. A ( unital) ring, described in the language of universal algebra, is an ordered tuple (R,+,\times,a,0,1) satisfying certain axioms, where + and \times are binary functions on the set R, a is a unary operation corresponding to additive inverse, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact Hausdorff Space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which are of particular interest in mathematical analysis. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CW Complexes
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. (open access) CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation (often with a much smaller complex). The C in CW stands for "closure-finite", and the W for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces \emptyset = X_ \subset X_0 \subset X_1 \subset \cdots such that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to the open k-ball B^k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internal Hom
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. Formal definition Let ''C'' be a locally small category (i.e. a category (mathematics), category for which hom-classes are actually Set (mathematics), sets and not proper classes). For all objects ''A'' and ''B'' in ''C'' we define two functors to the category of sets as follows: : The functor Hom(–, ''B'') is also called the ''functor of points'' of the object ''B''. Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms. The pair of functors Hom(''A'', –) and Hom(–, ''B'') are related in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
