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
Consider the mapping space of those homeomorphisms from the closed ball to itself which restrict to the identity on the boundary -sphere. What is known as Alexander’s trick is a construction (due to Alexander 1923) of an explicit deformation which witnesses that this space is connected, hence that any two such homemorphisms are connected by an isotopy.
Named after:
See also:
Wikipedia, Alexander’s trick
Created on September 20, 2022 at 16:38:36. See the history of this page for a list of all contributions to it.