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
A homotopy between continuous functions between topological spaces is called delayed if it starts out being constant near one boundary of the interval.
(If it is constant near both boundaries we say it has sitting instants).
For the unit interval and and any topological spaces, a continuous map is a delayed homotopy (between and if there exist such that for all .
Delayed homotopies appear in an alternative characterization of Dold fibrations. See there for details.
If a continuous homotopy between two smooth functions is delayed at both ends of the inerval it may be approximated by a smooth homotopy . See Steenrod-Wockel approximation theorem.
Last revised on October 25, 2010 at 17:16:55. See the history of this page for a list of all contributions to it.