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 closed cover of a topological space $X$ is a collection $\{U_i \subset X\}$ of closed subsets of $X$ whose union equals $X$: $\cup_i U_i = X$.
Often it is also required that every point $x \in X$ is in the interior of one of the $U_i$ (e.g in Floyd 1957, but e.g. not in Karoubi & Weibel 2016).
Closed covers can be obtained from open covers by forming the closure of each of the open subsets. The result clearly satisfies the clause that every point is in the interior of one of the closed subsets.
Dragan Janković, Chariklia Konstadilaki, On covering properties by regular closed sets, Mathematica Pannonica, 7/1 (1996) 97-111 [pdf]
Max Karoubi, Charles Weibel, On the covering type of a space, L’Enseignement Mathématique, 62 3/4 (2016) 457-474 [arXiv:1612.00532, doi:10.4171/LEM/62-3/4-4]
Applications of closed covers in Čech homology:
Related discussion is also in this MO thread
In analytic geometry, affinoid domains have closed sets as analytic spectra and hence the topological space underlying a Berkovich analytic spaces is equipped by a closed cover by affioid domains
Last revised on June 29, 2022 at 19:42:50. See the history of this page for a list of all contributions to it.