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
(Michael's theorem, 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.
Note that while in Michael’s paper he assumes that paracompact spaces are Hausdorff, the assumption is not necessary, and the proof goes through for regular non-Hausdorff spaces. See Kelley, p. 156.
(second-countable regular spaces are paracompact)
Let $X$ be a topological space which is
Then $X$ is paracompact topological space.
Let $\{U_i \subset X\}_{i \in I}$ be an open cover. By Michael's theorem (lemma ) it is sufficient that we find a refinement by a countable cover.
But second countability implies precisely that every open cover has a countable subcover:
Every open cover has a refinement by a cover consisting of base elements, and if there is only a countable set of these, then the resulting refinement necessarily contains at most this countable set of distinct open subsets.
Kelley, General Topology, 1955.
Ernest Michael, A note on paracompact spaces, Proc. Amer. Math. Soc. 1953 (jstor, pdf)
Last revised on May 19, 2020 at 01:25:19. See the history of this page for a list of all contributions to it.