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
(locally compact topological space)
A topological space $X$ is called locally compact if for every point $x \in X$ and every open neighbourhood $U_x \supset \{x\}$ there exists a smaller open neighbourhood $V_x \subset U_x$ whose topological closure is compact and still contained in $U$:
(proper maps to locally compact spaces are closed)
Let
$(X,\tau_X)$ be a topological space,
$(Y,\tau_Y)$ a locally compact topological space according to def. 1,
$f \colon X \to Y$ a continuous function.
Then:
If $f$ is a proper map, then it is a closed map.
Let $C \subset X$ be a closed subset. We need to show that every $y \in Y \backslash f(C)$ has an open neighbourhood $U_y \supset \{y\}$ not intersecting $f(C)$ (by this prop.).
By local compactness of $(Y,\tau_Y)$ (def. 1), $y$ has an open neighbourhood $V_y$ whose topological closure $Cl(V_y)$ is compact. Hence since $f$ is proper, also $f^{-1}(Cl(V_y)) \subset X$ is compact. Then also the intersection $C \cap f^{-1}(Cl(V_y))$ is compact, and hence so is
This is also a closed subset, since compact subspaces of Hausdorff spaces are closed. Therefore
is an open neighbourhod of $y$ not intersecting $f(C)$.
injective proper maps to locally compact spaces are equivalently the closed embeddings
maps from compact spaces to Hausdorff spaces are closed and proper
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity