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
Stably compact spaces are topological spaces which share many of the desirable properties of compact Hausdorff spaces, such as compactness and local compactness, without being Hausdorff or even T1.
They are also a convenient setting for convergence in an ordered setting, being deeply linked to compact ordered spaces.
A topological space is called stably compact if the following conditions are met:
Note that the latter notion of coherence is slightly different than the one given at coherent space.
(…)
(…) For details see (Jung-Moshier 2006).
Achim Jung, Stably compact spaces and the probabilistic powerspace construction, ENTCS 87, 2004 (doi:10.1016/j.entcs.2004.10.001).
Achim Jung and M. Andrew Moshier. On the bitopological nature of Stone duality. Technical Report CSR-06-13. School of Computer Science, University of Birmingham, December 2006, 110 pages. pdf
Last revised on May 18, 2022 at 16:48:51. See the history of this page for a list of all contributions to it.