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
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
The action of a topological group $G$ on a topological space $X$ is called properly discontinuous if every point $x \in X$ has a neighbourhood $U_x$ such that the intersection $g(U_x) \cap U_x$ with its translate under the group action via some element $g \in G$ is non-empty only for the neutral element $e \in G$:
This is equivalent to the condition that the quotient space coprojection $X \longrightarrow X/G$ is a covering space-projection.
Therefore properly discontinuous actions are also called covering space actions (Hatcher).
Jack Lee, chapter 12 of Introduction to Topological Manifolds
Jack Lee, chapter 21 of Introduction to Smooth Manifolds
Jack Lee, MO comment, Dec. 2014
Created on April 24, 2018 at 13:05:27. See the history of this page for a list of all contributions to it.