This page is about the concept in topology. For the more general concept see at closed morphism.
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
(open maps and closed maps)
A continuous function is called
an open map if the image under of an open subset of is an open subset of ;
a closed map if the image under of a closed subset of (def. ) is a closed subset of .
(image projections of open/closed maps are themselves open/closed)
If a continuous function is an open map or closed map (def. ) then so its its image projection , respectively, for regarded with its subspace topology.
If is an open map, and is an open subset, so that is also open in , then, since , it is also still open in the subspace topology, hence is an open map.
If is a closed map, and is a closed subset so that also is a closed subset, then the complement is open in and hence is open in the subspace topology, which means that is closed in the subspace topology.
(maps from compact spaces to Hausdorff spaces are closed and proper)
Let be a continuous function between topological spaces such that
Then is
a closed map (def. );
a proper map.
(proper maps to locally compact spaces are closed)
Let
be a topological space,
a locally compact topological space according to def. ,
Then:
If is a proper map, then it is a closed map.
Last revised on May 12, 2017 at 21:38:31. See the history of this page for a list of all contributions to it.