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 generalization of the notion of topological spaces with proper maps between them, resulting in a bicomplete category.
An exterior space is a topological space equipped with an externology: a collection of open subsets (known as exterior sets) that is closed under finite intersection and passage to open supersets.
A morphism of exterior spaces is a continuous map such that the preimage of an exterior set is exterior.
The category of exterior spaces is complete and cocomplete.
The category of topological spaces and proper maps embeds fully faithfully into exterior spaces.
Thus, exterior spaces form a natural categorical setting for proper homotopy theory.
Original reference:
A modern source with references to earlier articles:
Last revised on August 8, 2026 at 16:51:40. See the history of this page for a list of all contributions to it.