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
A topological space is called extremally disconnected if the closure of any open subset is still an open subset.
Extremally disconnected topological spaces are precisely the projective objects in the category of compact Hausdorff topological spaces.
See e.g. (Bhatt-Scholze 13, below theorem 1.8)
For $R$ a w-contractible ring, the profinite set $\pi_0(Spec R)$ is an extremally disconnected profinite set.
Part of (Bhatt-Scholze 13, theorem 1.8).
The extremal disconnectedness of a space is correlated with the property that the frame of its open subsets is a De Morgan Heyting algebra hence with the validity of the De Morgan law in logic, since a result by Johnstone says that a topos $Sh(X)$ of sheaves on a space $X$ is a De Morgan topos precisely when $X$ is extremally disconnected and this implies that all subobject lattices $sub(X)$ are De Morgan Heyting algebras, in particular $sub(1)$ corresponding to the frame of opens of $X$ (cf. De Morgan topos for details and references).
For $S$ any set regarded as a discrete topological space, its Stone-Cech compactification is extremally disconnected.
(e.g. Bhatt-Scholze 13, example 2.4.6)
The profinite set underlying the p-adic integers, regarded as a Stone space, is not extremally disconnected.
(e.g. Bhatt-Scholze 13, example 2.4.7)
Discussion in the context of the pro-etale site is in
The result on projective spaces stems from
Last revised on January 3, 2021 at 12:43:14. See the history of this page for a list of all contributions to it.