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 locale is paracompact if it is regular and every open cover has a locally finite refinement.
Paracompact locales are very closely related to fully normal locales?. In fact, for regular locales these two properties are equivalent.
Any metrizable locale? is paracompact.
Any Lindelöf locale? is paracompact.
A locale is paracompact if and only if it admits a complete uniformity.
The full subcategory of paracompact locales is a reflective subcategory of the category of completely regular locales as well as the category of all locales.
In particular, the inclusion functor from paracompact locales to locales preserves small limits, so in particular, products of paracompact locales are paracompact.
This last property clearly distinguishes paracompact locales from paracompact spaces, since products of paracompact spaces need not be paracompact.
fully normal locale?
Last revised on May 28, 2026 at 12:46:09. See the history of this page for a list of all contributions to it.