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 locale is locally compact if its frame of opens is a continuous poset.
In classical mathematics, every locally compact locale is spatial, hence may be regarded as a locally compact topological space.
Prop. may fail in constructive mathematics. For instance any spectrum of a commutative ring, considered as a locale, is locally compact, but may fail to be spatial (because of the lack of filters).
Locally compact locales are also exactly the exponentiable objects in the category of locales.
Last revised on June 19, 2022 at 17:24:24. See the history of this page for a list of all contributions to it.