Exercise 2, § 2

a) Extend Propositions 1, 2 and 4 to semi-topological groups (§ 1, Exercise 2). (To prove that the closure $\overline{H}$ of a subgroup $H$ is a subgroup start by remarking that $SH \subset \overline{H}$ for all $s \in H$).

b) Let $G$ be the group $\mathbf{Z}^{(\mathbf{N})}$, and let $e_i$ ($i \in \mathbf{N}$) be the element of $G$ all of whose coordinates are zero except the $i$th, and whose $i$th coordinate is 1. For each $n \in \mathbf{N}$, let $V_n$ be the set of all $z = (z_k) \in G$ such that $z = 0$ for $k \leq n$ and $z_k \geq 0$ for $k > n$. The $V_n$ form a fundamental system of neighbourhoods of $0$ for a topology which makes $G$ into a Hausdorff paratopological group. Let $H$ be the subgroup of $G$ generated by the elements $e_0 + e_n \ (n \geq 1)$ Show that $H$ is discrete but not closed in $G$, and that $\overline{H}$ is not a subgroup of $G$.