nLab
properly discontinuous action

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Representation theory

Contents

Definition

The action of a topological group GG on a topological space XX is called properly discontinuous if every point xXx \in X has a neighbourhood U xU_x such that the intersection g(U x)U xg(U_x) \cap U_x with its translate under the group action via some element gGg \in G is non-empty only for the neutral element eGe \in G:

g(U x)U xAAAAg=e g(U_x) \cap U_x \neq \emptyset \phantom{AA} \Rightarrow \phantom{AA} g = e

This is equivalent to the condition that the quotient space coprojection XX/GX \longrightarrow X/G is a covering space-projection.

Therefore properly discontinuous actions are also called covering space actions (Hatcher).

References

Created on April 24, 2018 at 09:05:27. See the history of this page for a list of all contributions to it.