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 following terminology is used in topology:
(saturated subset)
Let be a function of sets. Then a subset is called an -saturated subset (or just saturated subset, if is understood) if is the pre-image of its image:
If is not necessarily -saturated, then is called its saturation. Notice that .
Let be a function. Then a subset is -saturated (def. ) precisely if its complement is so.
whose underlying function is surjective exhibits as the corresponding quotient topology precisely if sends open and -saturated subsets in (def. ) to open subsets of . By lemma this is the case precisely if it sends closed and -saturated subsets to closed subsets.
Let
be a closed map.
be a closed subset which is -saturated;
an open subset containing
then there exists a smaller open subset still containing
and such that is -saturated.
We claim that
has the desired properties. To see this, observe first that
the complement is closed, since is assumed to be open;
hence the image is closed, since is assumed to be a closed map;
hence the pre-image is closed, since is continuous, therefore its complement is indeed open;
this pre-image is saturated, as all pre-images clearly are, and hence also its complement is saturated, by lemma .
Therefore it now only remains to see that .
The inclusion means equivalently that , which is clearly the case.
The inclusion meas that . Since is saturated by assumption, this means that . This in turn holds precisely if . Since is saturated, this holds precisely if , and this is true by the assumption that .
Last revised on May 21, 2017 at 18:14:24. See the history of this page for a list of all contributions to it.