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 Stonean locale is an extremally disconnected Stone locale. The category of Stonean locales has open maps of locales as morphisms.
Thus, the category of Stonean locales is a (nonfull) subcategory of the category of Stone locales.
In presence of the axiom of choice, every Stonean locale is spatial and the category of Stonean locales is equivalent to the cateogry of Stonean spaces.
As a variant of Stone duality, the category of Stonean locales is contravariantly equivalent to the category of complete Boolean algebras and continuous homomorphisms. One can also reformulate this statement as an equivalence between the categories of Stonean locales and Boolean locales.
Unlike the corresponding statement for Stonean spaces, this version is fully constructive and is valid in any W-topos.
In fact, the traditional Stonean duality? is an immediate consequence of the localic Stonean duality and the spatiality of Stonean locales.
The definition is due to Marshall Stone, see
\bibitem{ACSBR} Marshall~H.~Stone, Algebraic characterizations of special Boolean rings. Fundamenta Mathematicae 29:1 (1937), 223–303.
Further investigations of Stonean and hyperstonean? spaces were done by Dixmier:
\bibitem{HS} Jacques Dixmier, Sur certains espaces considérés par M.~H.~Stone. Summa Brasiliensis Mathematicae 2 (1951), 151–182. PDF.
An expository account is available in Takesaki’s book
\bibitem{TOA} Masamichi Takesaki, Theory of operator algebras I. Encyclopaedia of Mathematical Sciences 124 (2002).
Last revised on July 24, 2019 at 00:59:50. See the history of this page for a list of all contributions to it.