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
Given a topological space , the category of open subsets of is the category whose
objects are the open subsets of ;
morphisms are the inclusions
of open subsets into each other.
The category is a poset, in fact a frame (dually a locale): it is the frame of opens of .
The category is naturally equipped with the structure of a site, where a collection of morphisms is a cover precisely if their union in equals :
The category of sheaves on equipped with this site structure is typically referred to as the category of sheaves on the topological space and denoted
The category is also a suplattice.
Last revised on November 5, 2023 at 21:28:11. See the history of this page for a list of all contributions to it.