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
The Arens-Fort space is the set of pairs of natural numbers with all subsets not containing being open and all subsets containing and containing almost all (all but finitely many) points in almosts all columns being open.
(E.g. Joshi 83, Chapter 4, Section 2, Example 10)
The Arens-Fort space is:
The Arens-Forst space is not:
See also:
Last revised on June 1, 2024 at 04:15:10. See the history of this page for a list of all contributions to it.