topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
A topological space $X$ is weakly Hausdorff (or weak Hausdorff) if for any compact Hausdorff space $K$ and every continuous map $f\colon K\to X$, the image $f(K)$ is closed. Every weakly Hausdorff space is $T_1$ (that is every point is closed), and every Hausdorff space is weakly Hausdorff. For the most common purposes for which Hausdorff spaces are used, the assumption of being weakly Hausdorff suffices. See also compactly generated space.
The notion also also makes sense as stated for locales. Where their theory overlaps (in sober spaces and topological locales), the notions of weak Hausdorffness agree, given the ultrafilter theorem (which implies that all compact Hausdorff spaces/locales are sober/topological).
(this is a left adjoint …) See for now (Strickland 2009, Proposition 2.22), for the weak Hausdorffification of a compactly-generated topological space. Note in particular that the construction of the weak Hausdorffification of a compactly-generated space is a quotient by a closed equivalence relation given by a one-step construction (the intersection of all closed equivalence relations on the space), as opposed to the equivalence relation generated by a possibly transfinite procedure as in the Hausdorffification of an arbitrary topological space.
Write CGWH for the category of compactly generated weakly Hausdorff topological spaces, and $CGH$ for compactly generated Hausdorff topological spaces. Both are convenient categories of topological spaces that both admit a homotopy hypothesis-comparison to simplicial sets, but CGWH has a key further property:
The construction of pushouts is better behaved in CGWH than in CGH. Specifically, CHWH is closed under pushouts, one leg of which is the inclusion of a closed subspace. CGH does not have such nice behavior, and pushouts like that are used all over The Geometry of Iterated Loop Spaces, specifically in the construction of a monad from an operad and in the use of geometric realizations of simplicial spaces.
(Peter May, MO answer, April 2015)
The category of compactly generated weakly Hausdorff topological spaces was introduced in
as a more convenient setting than Steenrod’s compactly generated Hausdorff spaces, given that the latter is not closed under many colimits (for instance quotients) as computed in $Top$.
A survey is in
Last revised on May 25, 2023 at 01:51:49. See the history of this page for a list of all contributions to it.