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 family of subsets of a topological space, for which for every point the intersection of subsets containing it is open, is called an inner-preserving family or Q family.
A topological space, for which every (countable) open covering has an inner-preserving open refinement, is called (countably) orthocompact.
Orthocompactness generalizes other notations of compactness:
Closed subsets of (countably) orthocompact spaces are (countably) orthocompact.
Orthocompact spaces are countably orthocompact and countably orthocompact Lindelöf spaces are orthocompact.
Follows directly from the fact that every open cover of a Lindelöf space has a countable sub-cover.
P-spaces are countably orthocompact.
Follows directly from the fact that every countable open cover of a P-space is already inner-preserving.
Spaces with Alexandroff topology are orthocompact.
Follows directly from the fact that every open cover of a space with Alexandroff topology is already inner-preserving.
Given an orthocompact space $X$, the product space $X\times[0,1]$ is orthocompact iff $X$ is countably metacompact.
(Scott 75)
See also:
Last revised on June 10, 2024 at 11:24:27. See the history of this page for a list of all contributions to it.