nLab
almost connected topological group

Context

Group Theory

Topology

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

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 include

  • 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 include

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

Revised on February 3, 2012 22:02:40 by Todd Trimble (74.88.146.52)