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
Selection theory asks if given a multi-valued function $F\colon X \to Y$ does there exist a continuous selector $f\colon X \to Y$, i.e. a single-valued continuous function such that $f(x) \in F(x)$ for all $x \in X$. A selection theorem states that under certain assumptions on $X, Y, F$ there is indeed a selector. Normally, such assumptions include that $X$ is paracompact, $Y$ is some subset of a topological vector space, $F$ is a lower semicontinuous map (also called hemicontinuous), and $F(x)$ is convex for each $x \in X$.
More generally, one may ask if there is a multi-valued function $G\colon X \to Y$ such that $G(x) \subset F(y)$ for all $y\in X$ and $G$ nicer behaved than $F$, e.g. $G$ lower semicontinuous and $G(x)$ compact for every $x\in G$ or $G$ single-valued and measurable.
[Michael selection theorem] Let $X$ be paracompact, $Y$ a Banach space, $F$ a lower semicontinuous map, and $F(x)$ nonempty, convex, and closed for every $x\in X$. Then $F$ admits a selector.
Michael selection theorem appeared in
Overviews of selection theorems is found in
See also