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 $U$:
(open subspaces of compact Hausdorff spaces are locally compact)
Every open topological subspace $X \underset{\text{open}}{\subset} K$ of a compact Hausdorff space $K$ is a locally compact topological space.
In particular every compact Hausdorff space itself is locally compact.
Let $x \in X$ be a point and let $U_x \subset X$ an open neighbourhood. We need to produce a small open neighbourhood whose closure is compact and still contained in $U_x$.
By the nature of the subspace topology there exists an open subset $V_x\subset K$ such that $U_x = X \cap V_x$. Since $X$ is assumed to be open, it follows that $U$ is also open as a subset of $K$. Since compact Hausdorff spaces are normal it follows (by this prop.) that there exists a smaller open neighbourhood $W_x \subset K$ whose topological closure is still contained in $U_x$, and since closed subspaces of compact spaces are compact:
The intersection of this situation with $X$ is the required smaller compact neighbourhood $Cl(W_x) \cap X$:
Conversely, every locally compact Haudorff space $X$ arises as in prop. , since it may be considered an open subspace in its one-point compactification $X \sqcup \{\infty\}$. See there this example.