nLab
almost connected topological group

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.

Related concepts

• connected topological space

• maximal compact subgroup

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)