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
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
A condensed object in the category Ab of abelian groups.
The category of condensed abelian groups enjoys excellent categorical properties for homological algebra:
It is an abelian category that admits all small limits and colimits;
In this category, filtered colimits and infinite products are exact. The latter property is rather rare.
It has enough compact projective objects: free condensed abelian groups on extremally disconnected compact Hausdorff topological spaces generate all condensed abelian groups under small colimits and their corepresentable functors reflect isomorphisms;
The previous property implies that condensed abelian groups have the same exactness properties as the category of abelian groups.
Last revised on May 29, 2022 at 22:52:17. See the history of this page for a list of all contributions to it.