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
There are several theorems of Ernest Michael from the 1950s about paracompactness. There is also Michael’s 1972 characterization of paracompact locally compact spaces under certain class of quotient maps.
(detection of paracompactness, Michael 53, theorem 1)
Let $X$ be a topological space such that
$X$ is regular;
every open cover of $X$ has a refinement by a union of a countable set of
locally finite sets of open subsets (not necessarily covering).
Then $X$ is paracompact topological space.
Prop. immediately implies that
(on the closed image of a paracompact space, Michael 57, corollary 1)
The image of a paracompact Hausdorff space under a closed continuous function is also paracompact Hausdorff.
(Michael selection theorem)
A lower semicontinuous map from a paracompact topological space $X$ to a Banach space $E$ with convex closed values has a continuous subrelation which is a function. If this is true for a given topological space $Y$ instead of $E$ and all such functions and codomains $E$, then $Y$ is paracompact.
The original articles are the following:
Ernest Michael, A note on paracompact spaces, Proc. Amer. Math. Soc. 1953 (jstor, pdf)
Ernest Michael, Another note on paracompactness, 1957, (pdf, pdf)
Ernest Michael, Continuous selections. I, Annals of Math. 63 (2): 361–382 (doi:10.2307/1969615, MR0077107)
Ernest Michael, A quintuple quotient test, Gen. Topol. Appl. 2 (1972) 91–138, link
discussion at A. Mathew’s blog and also on an application to metric spaces here
See also:
R. Engelking, General topology
Kenneth Kunen, Paracompactness of box products of compact spaces, Trans. Amer. Math. Soc. 240, (1978) 307-316, jstor
wikipedia Michael selection theorem
Last revised on June 1, 2020 at 11:43:25. See the history of this page for a list of all contributions to it.