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 called hemicompact iff there exists a sequence $(K_n)_{n\in\mathbb{N}}$ of compact subsets $K_n\subseteq X$ such that $X=\bigcup_{n\in\mathbb{N}} K_n$ and $K_n \subseteq int(K_{n+1})$ holds.
If $X$ is hemicompact and $(K_n)$ a sequence as in the definition, then every compact subset $K\subseteq X$ is contained in one of the $K_n$. In other words: $\lbrace K_n \mid n\in\mathbb{N}\rbrace$ is cofinal in the set of all compact subsets (partially ordered by inclusion)
Locally compact (every point has a compact neighborhood) and $\sigma$-compact spaces are hemicompact (and also paracompact).
A first-countable and hemicompact space is locally compact (every point has a compact neighborhood).
Here the condition to be first-countable can’t be dropped as there are hemicompact spaces, which are not locally compact. An example is the Arens-Fort space as seen in Joshi 83, Chapter 4, Section 2, Example 10, which is not even compact.
A hemicompact space is $\sigma$-compact.
The field of rational numbers $\mathbb{Q}$ is $\sigma$-compact, but not hemicompact.
(Willard 04, p. 126, answer by Eric Wofsey on MSE/4209303)
See also:
Last revised on June 10, 2024 at 11:24:08. See the history of this page for a list of all contributions to it.