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 Stonean space is a compact, Hausdorff extremally disconnected topological space. Stonean spaces form a category if we take continuous open maps as morphisms.
Thus, the category of Stonean spaces is a (nonfull) subcategory of the category of Stone spaces and continuous maps.
In presence of the axiom of choice, as a consequence of Stone duality, the category of Stonean spaces is contravariantly equivalent to the category of complete Boolean algebras and continuous homomorphisms. This statement is known as the Stonean duality.
Without the axiom of choice, the category of Stonean locales is contravariantly equivalent to the category of complete Boolean algebras and continuous homomorphisms. See the article Stonean locale for more information.
A standard textbook is
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