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 topological subspace is a neighborhood retract of a topological space if there is a neighborhood in such that is a retract of .
A metrisable topological space is an absolute neighborhood retract if for any embedding as a closed subspace in a metrisable topological space , is a neighborhood retract of .
A pair where is a closed subspace of is an NDR-pair (for ‘Neighbourhood Deformation Retract’) or a closed Borsuk pair if there is a function and a homotopy such that , for all , for all , for all such that and . (See deformation retract.)
The canonical inclusion corresponding to any NDR-pair is a closed Hurewicz cofibration.
Textbook accounts
Last revised on October 19, 2023 at 01:23:11. See the history of this page for a list of all contributions to it.