Exercise 1, § 1

Let $X$ be a set, let $Y$ be a uniform space containing more than one point, and let $\mathscr{S}$ be a non-empty set of non-empty subsets of $X$. Let $Y' \subset \mathscr{F}(X; Y)$ be the set of all constant mappings of $X$ into $Y$. For each $y \in Y$, let $c_y$ be the constant mapping of $X$ into $Y$ whose value is $y$.

a) Show that $y \to c_y$ is an isomorphism of $Y$ onto the uniform subspace $Y'$ of $\mathscr{F}_{\mathscr{S}}(X; Y)$.

b) $\mathscr{F}_{\mathscr{S}}(X; Y)$ is Hausdorff if and only if $Y$ is Hausdorff and $\mathscr{S}$ is a covering of $X$.

c) $Y'$ is closed in $\mathscr{F}_{\mathscr{S}}(X; Y)$ if and only if $\mathscr{F}_{\mathscr{S}}(X; Y)$ is Hausdorff.