Exercise 18, § 4

Let $X$ be a replete space.

(a) Now prove that for every non-identically zero homomorphism $u$ of the algebra $C(X; \mathbf{R})$ into $\mathbf{R}$, there exists one and only one point $x \in X$ such that $u(f) = f(x)$ for every $f \in C(X; \mathbf{R})$.

(b) Let $Y$ be a second replete space. Now prove that for every homomorphism of $\mathbf{R}$-algebras $v : C(Y; \mathbf{R}) \to C(X; \mathbf{R})$ transforming the unit element into a unit element, there exists one and only one continuous mapping $\omega : X \to Y$ such that $v(g) = g \circ w$ (use (a)). In particular, for $C(X; \mathbf{R})$ and $C(Y; \mathbf{R})$ to be isomorphic, it is necessary and sufficient that $X$ and $Y$ be homeomorphic.