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 Stone space is a compact, Hausdorff totally disconnected topological space.
Stone spaces are sometimes called profinite spaces, since they are precisely the spaces which are small cofiltered limits of finite discrete spaces, and moreover (as a consequence of Stone duality) the category of Stone spaces is equivalent to the category of pro-objects in FinSet and finite sets sit as finite discrete spaces. This is especially common when talking about profinite groups and related topics.
A standard textbook is
See also
See also
S. B. Niefield, K.I. Rosenthal?, Sheaves of integral domains on stone spaces, Journal of Pure and Applied Algebra Volume 47, Issue 2, August 1987, Pages 173–179
Felix Cherubini, Thierry Coquand, Freek Geerligs, Hugo Moeneclaey, A Foundation for Synthetic Stone Duality (arXiv:2412.03203)
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