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
higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
A localic groupoid is a internal groupoid in the category of locales. A special case is of localic groups.
Localic groupoids are important, among other reasons, because every Grothendieck topos can be presented as the topos of equivariant sheaves on some localic groupoid. This fact is due to Joyal and Tierney. For more see classifying topos of a localic groupoid.
localic infinity-groupoid?
The Joyal–Tierney theorem appeared in
An expository account of the Joyal–Tierney theorem:
Last revised on May 25, 2023 at 13:39:35. See the history of this page for a list of all contributions to it.