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 Stone locale is a compact zero-dimensional? locale.
In presence of the axiom of choice, every Stone locale is spatial? and the category of Stone locales is equivalent to the category of Stone spaces.
The Stone duality theorem states that the category of Stone locales is contravariantly equivalent to the category of Boolean algebras.
Unlike the corresponding statement for Stone spaces, this version is fully constructive and is valid in any W-topos.
In fact, the traditional Stone duality is an immediate consequence of the localic Stone duality and the spatiality of Stone locales.
Created on July 23, 2019 at 23:24:28. See the history of this page for a list of all contributions to it.