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
h
symmetric monoidal (∞,1)-category of spectra
The topological space underlying a scheme is a sober topological space. In particular for affine schemes: The Zariski topology on the prime spectrum of a commutative ring is sober.
For proof see this prop at Zariski topology.
If instead of the prime spectrum of a commutative ring one considers only the topological subspace of maximal ideals in prop. 1, as in algebraic geometry before the introduction of schemes by Alexander Grothendieck, then one does not get a sober space. But if the ring is a Jacobson ring (in that every prime ideal in the ring is an intersection of maximal ideals), then the soberification of the topological subspace of maximal ideals inside the prime ideals is the full prime spectrum.