Exercise 9, § 1

a) Let $X$ be a topological space, let $\mathscr{T}$ be its topology and let $(f_i)_{i \in I}$ be a family of mappings of $X$ into $K = [0, 1]$ which are continuous in the topology $\mathscr{T}$. Let $\mathscr{T}_0$ be the coarsest topology on $X$ for which all the $f_i$ are continuous. Show that $\mathscr{T}_0 = \mathscr{T}$ provided that the family of sets $\overline{f_i}([0, a[)$, where $i \in I$ and $a \in ]0, 1[$, is a subbase of $\mathscr{T}$ (Chapter I, § 2, no. 3). The space $X$ is then uniformizable, and if it is Hausdorff it is homeomorphic to a subspace of the cube $K^I$.

b) Suppose that the family $(f_i)$ is the family of all continuous mappings of $X$ into $K$ (with respect to $\mathscr{T}$). Show that, if $Y$ is any compact space, every mapping of $X$ into $Y$ which is continuous with respect to $\mathscr{T}$ is continuous with respect to $\mathscr{T}_0$ (embed $Y$ in a cube). The topology $\mathscr{T}$ is uniformizable if and only if $\mathscr{T}_0 = \mathscr{T}$.