nLab almost connected topological group

Contents

Context

Group Theory

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

Contents

Definition

Definition

A locally compact topological group GG is called almost connected if the underlying topological space of the quotient topological group G/G 0G/G_0 (of GG by the connected component of the neutral element, also called the identity component) is compact.

See for instance (Hofmann-Morris, def. 4.24).

Remark

Since the connected component G 0G_0 is closed, the neutral element in G/G 0G/G_0 is a closed point. It follows that G/G 0G/G_0 is T 1 T_1 and therefore (because it is a uniform space) also T 312T_{3 \frac1{2}} (a Tychonoff space; see at uniform space for details). In particular, G/G 0G/G_0 is a compact Hausdorff space.

Example

Every compact and every connected topological group is almost connected.

Also every quotient of an almost connected group is almost connected.

References

Textbooks with relevant material:

  • M. Stroppel, Locally compact groups, European Math. Soc., (2006)

  • Karl Hofmann Sidney Morris, The Lie theory of connected pro-Lie groups, Tracts in Mathematics 2, European Mathematical Society, (2000)

Original articles:

  • Chabert, Echterhoff, Nest, The Connes-Kasparov conjecture for almost connected groups and for linear pp-adic groups (pdf)

Last revised on July 8, 2022 at 12:42:10. See the history of this page for a list of all contributions to it.