# nLab almost connected topological group

group theory

### Cohomology and Extensions

#### Topology

topology

algebraic topology

# Contents

## Definition

###### Definition

A locally compact topological group $G$ is called almost connected if the underlying topological space of the quotient topological group $G/{G}_{0}$ (of $G$ 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 ${G}_{0}$ is closed, the identity in $G/{G}_{0}$ is a closed point. It follows that $G/{G}_{0}$ is ${T}_{1}$ and therefore, because it is a uniform space, ${T}_{3\frac{1}{2}}$ (a Tychonoff space; see uniform space for details). In particular, $G/{G}_{0}$ 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 $p$-adic groups (pdf)

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