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 coherent if its compact elements are closed under finite meets and any element is a join of compact elements.
A morphism of coherent locales is a morphism of locales such that preserves compact elements.
Every coherent locale is a coherent topos which is also a locale (i.e. (0,1)-topos).
Coherent locales are spatial. This is a special case of the Deligne completeness theorem for coherent toposes. The corresponding topological spaces are called coherent spaces.
Coherent locales classify coherent theories. If is a propositional coherent theory, then the model is a coherent locale.
The category of coherent locales is contravariantly equivalent to the category of distributive lattices.
See Exercise 24 of:
Created on April 8, 2025 at 14:45:23. See the history of this page for a list of all contributions to it.