Exercise 18, § 4

Let $X$ be a real-compact space.

a) Show that, for each non-zero homomorphism $u : C(X; \mathbf{R}) \to \mathbf{R}$, there exists a unique point $x \in X$ such that $u(f) = f(x)$ for all $f$ in $C(X; \mathbf{R})$.

b) Let $Y$ be another real-compact space. Show that, for every $\mathbf{R}$-algebra homomorphism $v : C(Y; \mathbf{R}) \to C(X; \mathbf{R})$ which maps identity element to identity element, there exists a unique continuous mapping $w : X \to Y$ such that $v(g) = g \circ w$ for all $g \in C(Y; \mathbf{R})$. In particular, $C(X; \mathbf{R})$ and $C(Y; \mathbf{R})$ are isomorphic if and only if $X$ and $Y$ are homeomorphic.