Exercise 11, § 1

Let X be a topological space and G a topological group. The set $\mathscr{C}(X; G)$ of continuous mappings of X into G is a subgroup of the group $G^X$. Let $\mathfrak{S}$ be a set of subsets of X.

a) Suppose that for each $A \in \mathscr{S}$, each neighbourhood $V$ of $e$ in $G$ and each $u \in \mathscr{C}(X; G)$, there exists a neighbourhood $W$ of $e$ in $G$ such that $sWs^{-1} \subset W$ for all $s \in u(A)$. Show that the topology of $\mathscr{S}$-convergence is then compatible with the group structure of $\mathscr{C}(X; G)$, and that the right (resp. left) uniformity of the topological group $\mathscr{C}_{\mathscr{S}}(X; G)$ so defined is the same as the uniformity of $\mathscr{S}$-convergence. Consider the case of pointwise convergence, the case of compact convergence when $G$ is locally compact, and the case where $G$ is abelian.

b) If $G$ is the group $\mathbf{SL}(2, \mathbf{R})$ endowed with the topology induced by that of $\mathbf{R}^4$, show that the topology of uniform convergence is not compatible with the group structure of $\mathscr{C}(\mathbf{R}; G)$.