nLab
almost connected topological group
Context
Group Theory
group theory

Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Topology
topology (point-set topology , point-free topology )

see also differential topology , algebraic topology , functional analysis and topological homotopy theory

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents
Definition
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, $T_{3 \frac1{2}}$ (a Tychonoff space ; see uniform space for details). In particular, $G/G_0$ is compact Hausdorff.

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 )