Exercise 12, § 4

(a) Let $X$ be a compact space, $f : X \to Y$ a continuous surjective mapping from $X$ onto a compact space $Y$. Now prove that the mapping $af : u \mapsto u \circ f$ from $C(Y; \mathbf{R})$ into $C(X; \mathbf{R})$ is an isomorphism (preserving the norm) of $C(Y; \mathbf{R})$ onto a closed subalgebra of $C(X; \mathbf{R})$ containing the unit element.

(b) Conversely, let $A$ be a closed subalgebra of $C(X; \mathbf{R})$ containing the unit element. Now prove that there exists a continuous surjective mapping $f$ from $X$ onto a compact space $Y$ such that $af$ is an isomorphism of $C(Y; \mathbf{R})$ onto $A$. (If $(u_\lambda)_{\lambda \in L}$ is a everywhere dense family in $A$, consider the continuous mapping $x \mapsto (u_\lambda(x))$ from $X$ into $\mathbf{R}^L$ and use the th. of Weierstrass-Stone).

(c) Deduce from b) that for every sequence $(u_n)$ of elements of $C(X; \mathbf{R})$, there exists a metrizable compact space $Y$ and a continuous surjective mapping $f : X \to Y$ such that the $u_n$ belong to the image of $C(Y; \mathbf{R})$ by $af$.