Exercise 8, § 1

Let X be a metrizable topological space; let $d$ be a distance which defines its topology. Let $\mathscr{C}_b(X;\mathbf{R})$ be the vector space of real-valued functions on X which are continuous and bounded; it is endowed with the norm defined by $\|f\|=$ sup$_{x\in X}|f(x)|$ for $f\in \mathscr{C}_b(X;\mathbf{R})$.

a) Define a mapping $j: X\rightarrow \mathscr{C}_b(X;\mathbf{R})$ by setting $j(x):y\mapsto$ $d(x, y)/(1 +d(x, y))$. Exercise prove that the mapping $j$ induces a homeomorphism of X onto its image.

b) Let V be the intersection of all convex subsets of $\mathscr{C}_b(X;\mathbf{R})$ which contain $j(X)$. Exercise prove that $j(X)$ is closed in V.

c) Let X be a metrizable space of countable type. Exercise prove that there exists a convex subset Z of $\mathbf{I}^{\mathbf{N}}$ such that X is homeomorphic to a closed subset of Z.