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
The relative version of a locally compact topological space is a continuous map that behaves like a ‘family of locally compact spaces’.
A continuous function $f \colon X \to Y$ is called locally proper if for any $x \in X$ and neighbourhood $V \ni x$ there is a neighbourhood $A\ni x$ with $A \subset V$ and a neighbourhood $U \ni f(x)$ such that $A \to U$ is a proper map.
The inclusion map of a locally closed subset $C\hookrightarrow X$ in a topological space $X$ is locally proper.
A topological space $X$ is a locally compact space if and only if $X\to \ast$ is locally proper.
Let $f\colon X\to Y$ and $g\colon Y\to Z$ be locally proper. Then $g\circ f$ is locally proper. If $W\to Y$ is any continuous function then $W\times_Y Z \to W$ is locally proper. (ie locally proper maps are closed under composition and stable under pullback. Hence they form a singleton coverage)
As a corollary, one has that for a locally compact space $X$, every locally closed subspace of $X$ is locally compact.
Let $f\colon X\to Y$ be a continuous function. Then if
or
then $f$ is locally proper. The converse implications also hold
Recall that a continuous map $g\colon Y \to Z$ of topological spaces is called separated if $Y\to Y\times_Z Y$ is a closed embedding.
If $f\colon X\to Y$ and $g\colon Y\to Z$ are continuous maps, $g$ is separated and $g\circ f$ is locally proper, then $f$ is locally proper
Every continuous map from a locally compact Hausdorff space to a Hausdorff space is separated and locally proper.
The following proposition generalises the well-known result that compact Hausdorff spaces are locally compact.
Every separated and proper map is locally proper
The proper base change theorem also holds for locally proper maps if one uses direct image with compact support instead of ordinary direct image
Last revised on February 5, 2020 at 11:39:19. See the history of this page for a list of all contributions to it.