In topology, a topological space 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 don't.
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 is open in if and only if is open in for every compact subspace Other definitions use a family of continuous maps from compact spaces to and declare to be compactly generated if its topology coincides with the final topology with respect to this family of maps. And other variations of the definition replace compact spaces with compact Hausdorff spaces.
Compactly generated spaces were developed to remedy some of the shortcomings of the category of topological spaces. In particular, under some of the definitions, they form a cartesian closed category while still containing the typical spaces of interest, which makes them convenient for use in algebraic topology.
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a topological space X {\displaystyle X} is called a compactlygeneratedspace or k-space if its topology is determined by compactspaces in a manner made...
mathematics, compactlygenerated can refer to: Compactlygenerated group, a topological group which is algebraically generated by one of its compact subsets...
mathematics, a compactlygenerated (topological) group is a topological group G which is algebraically generated by one of its compact subsets. This should...
In mathematics, the category of compactlygenerated weak Hausdorff spaces CGWH is one of typically used categories in algebraic topology as a substitute...
cube is compact, again a consequence of Tychonoff's theorem. A profinite group (e.g. Galois group) is compact. CompactlygeneratedspaceCompactness theorem...
compact. Quotient spaces of locally compact Hausdorff spaces are compactlygenerated. Conversely, every compactlygenerated Hausdorff space is a quotient...
with compactlygeneratedspaces as objects and continuous maps as morphisms or with the category of compactlygenerated weak Hausdorff spaces. Like many...
{\displaystyle (X,{\mathcal {T}})} is a compactlygeneratedspace, f n → f {\displaystyle f_{n}\to f} compactly, and each f n {\displaystyle f_{n}} is...
with compactlygeneratedspaces in algebraic topology. For that, see the category of compactlygenerated weak Hausdorff spaces. A k-Hausdorff space is a...
of non-compactness Paracompact space Locally compactspaceCompactlygeneratedspace Axiom of countability Sequential space First-countable space Second-countable...
difficulties in using homotopies with certain spaces. Algebraic topologists work with compactlygeneratedspaces, CW complexes, or spectra. Formally, a homotopy...
here. As an example, a commonly used variant of the notion of compactlygeneratedspace is defined as the final topology with respect to a proper class...
the change in functions More generally, on any compactlygeneratedspace; e.g., a first-countable space. Rudin 1991, p. 44 §2.5. Reed & Simon (1980), p...
non-compact, but this is not an open manifold since the circle (one of its components) is compact. Most books generally define a manifold as a space that...
"convenient". Every sequential space is compactlygenerated, and finite products in Seq coincide with those for compactlygeneratedspaces, since products in the...
that X and Y are compactlygenerated Hausdorff spaces, so Hom(X,Y) is often taken with the compactlygenerated variant of the compact-open topology; the...
topology, i.e., the topology generated by all intervals ( x , ∞ ) . {\displaystyle (x,\infty ).} The space is limit point compact because given any point a...
product and R as the unit. The category of pointed spaces (restricted to compactlygeneratedspaces for example) is monoidal with the smash product serving...
the partition. Finitely generatedspaces are those determined by the family of all finite subspaces. Compactlygeneratedspaces (in the sense of Definition...
setting of metric spaces. Other notions, such as continuity, compactness, and open and closed sets, can be defined for metric spaces, but also in the even...
theory for locally compactspaces from the functional-analytic viewpoint, it is necessary to extend measure (integral) from compactly supported continuous...
locally compact Polish space X {\displaystyle X} is a variant of the Vietoris topology, and is named after mathematician James Fell. It is generated by the...
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