Exercise 12, § 4

a) Let $X$ be a compact space, and let $f : X \to Y$ be a continuous surjection of $X$ onto a compact space $Y$. Show that the mapping

(*) This result does not extend to the case where $X$ is the non-metrizable compact space $\beta \mathbf{N}$ and $Y$ is the closed set $\beta \mathbf{N} - \mathbf{N}$.

af : u \to u \circ f of $\mathscr{C}(Y; \mathbf{R})$ into $\mathscr{C}(X; \mathbf{R})$ is a (norm-preserving) isomorphism of $\mathscr{C}(Y; \mathbf{R})$ onto a closed subalgebra of $\mathscr{C}(X; \mathbf{R})$ containing the identity element.

b) Conversely, let $A$ be a closed subalgebra of $\mathscr{C}(X; \mathbf{R})$ containing the identity element. Show that there exists a continuous surjection $f$ of $X$ onto a compact space $Y$ such that $af$ is an isomorphism of $\mathscr{C}(Y; \mathbf{R})$ onto $A$. {If $(u_\lambda)_{\lambda \in L}$ is a family which is dense in $A$, consider the continuous mapping $x \to (u_\lambda(x))$ of $X$ into $\mathbf{R}^L$ and use the Weierstrass-Stone theorem.}

c) Deduce from b) that, for each sequence $(u_n)$ of elements of $\mathscr{C}(X; \mathbf{R})$, 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 $\mathscr{C}(Y; \mathbf{R})$ under $af$.