Proposition 7

In a topological group $G$, the component $K$ of the identity element $e$ is a closed normal subgroup. The component of any point $x \in G$ is the coset $x.K = K.x$.

If $a \in K$, then $a^{-1}K$ is connected and contains $e$; hence $K^{-1}K \subset K$, which shows that $K$ is a subgroup of $G$. This subgroup is invariant under all automorphisms of $G$, and in particular under all inner automorphisms, therefore $K$ is normal in $G$; also $K$ is closed (Chapter I, , no. 5, Proposition 9). Finally, the left translation $y \to xy$ is a homeomorphism of $G$ which sends $e$ to $x$, and hence the component of $x$ is $x.K$

The component of the identity element $e$ of $G$ is called the identity component of $G$.