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
(one-point compactification of product space is smash product of the compactified factors)
On the subcategory $Top_{LCHaus}$ in Top of locally compact Hausdorff spaces with proper maps between them, the functor of one-point compactification (Prop. )
sends coproducts, hence disjoint union topological spaces, to wedge sums of pointed topological spaces;
sends Cartesian products, hence product topological spaces, to smash products of pointed topological spaces;
hence constitutes a strong monoidal functor for both monoidal structures of these distributive monoidal categories in that there are natural homeomorphisms
and
This is briefly mentioned in Bredon 93, p. 199. The argument is spelled out in: MO:a/1645794, Cutler 20, Prop. 1.6.
Basic accounts:
Review:
Last revised on January 8, 2021 at 07:52:06. See the history of this page for a list of all contributions to it.