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).

Remark

Since the connected component $G_0$ is closed, the neutral element 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) also $T_{3 \frac1{2}}$ (a Tychonoff space; see at uniform space for details). In particular, $G/G_0$ 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.

Related concepts

• connected topological space

• maximal compact subgroup

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 $p$-adic groups (pdf)