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
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
The analytic topology on an complex analytic space is the one given by covering the space by affine opens equipped with the standard topology induced from that of the complex numbers $\mathbb{C}^n$.
Chow's theorem states that a projective complex analytic variety, hence a closed analytic subspace of a complex projective space, is also an algebraic variety over the complex numbers. In general this is not true.
The comparison theorem (étale cohomology) relates the étale cohomology of a complex variety with the ordinary cohomology of its complex analytic topological space.
For more along such lines see at GAGA.
Brian Osserman, Complex varieties and the analytic topology (pdf)
James Milne, section 21 of Lectures on Étale Cohomology
Last revised on October 24, 2014 at 18:41:14. See the history of this page for a list of all contributions to it.