Identity component

Remarks

$G_0$ is a closed normal subgroup of $G$. It is a subgroup because $G_0 G_0^{-1}$ is the image of the connected set $G_0 \times G_0$ under $(x, y) \mapsto xy^{-1}$ and contains $e$, so it lies in $G_0$. It is normal because each inner automorphism $x \mapsto gxg^{-1}$ is a homeomorphism fixing $e$. The quotient $G/G_0$ is a totally disconnected group. Example: for $G = \mathrm{GL}_n(\R)$, the identity component is the subgroup of matrices with positive determinant. For $G = (\Q, +)$ with the topology from $\R$, the identity component is $\set{0}$, since $\Q$ is totally disconnected.