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 subset of a real affine space is convex if for any two points , the straight line segment connecting with in is also contained in . In other words, for any , and any , we have also .
Every convex set is star-shaped about each of its points, and hence contractible provided it is inhabited.
One generalization of convexity to Riemannian manifolds and metric spaces is geodesic convexity.
An abstract generalization of the notion of a convex set is that of a convex space. There is a nice characterization of those convex spaces which are isomorphic to convex subsets of real affine space.
The convex hull of a subset is the smallest convex subset containing it.
Last revised on November 15, 2023 at 04:21:56. See the history of this page for a list of all contributions to it.