Exercise 4, § 2

a) In a semi-topological group $G$, the intersection of the neighbourhoods of $e$ which are both open and closed is a closed subgroup $H$ which is invariant under all automorphisms of $G$ (remark that there can be no partition of $G$ into two open and closed subsets, each of which meets $H$).

b) On the set $\mathfrak{G}$ of all subgroups of $G$, define a mapping $\varphi : \mathfrak{G} \to \mathfrak{G}$ as follows: if $H$ is any subgroup of $G$, then $\varphi(H)$ is the subgroup of $H$ which is the intersection of all the simultaneously open and closed neighbourhoods of $e$ in $H$. In the set $\mathfrak{G}$, ordered by the relation $\supset$, consider the chain $\Gamma$ of $G$ with respect to the mapping $\varphi$ (Set Theory, Chapter III, § 2, Exercise 6); show that the smallest subgroup of $\Gamma$ is the connected component of $e$ in $G$ (remark that this smallest subgroup must be connected).