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 Boolean if all of its opens (i.e. elements of the corresponding frame) are complemented, i.e., for any we have . Equivalently, we could say that for all we have .
The underlying frames of Boolean locales are precisely complete Boolean algebras.
Maps of Boolean locales are automatically open. Their underlying morphisms of frames are precisely complete Boolean homomorphisms?, i.e., suprema-preserving homomorphisms of Boolean algebras.
Thus, the opposite category of Boolean locales is precisely the category of complete Boolean algebras and complete homomorphisms thereof.
By Stonean duality the category of Boolean locales is equivalent to the category of Stonean locales and open maps thereof. (Not every map of locales between Stonean locales is open, unlike for Boolean locales.)
Any locale has a maximal dense sublocale, whose opens are precisely the regular elements of , i.e., elements such that . This sublocale is Boolean and is also known as the double negation sublocale.
Last revised on July 24, 2019 at 00:50:18. See the history of this page for a list of all contributions to it.