Exercise 6, § 4

Let X be a completely regular space, and let $\mathscr{C}^b(X; \mathbf{R})$ be the normed subalgebra of $\mathscr{B}(X; \mathbf{R})$ formed by the continuous and bounded numerical functions in X. In order that every subalgebra of $\mathscr{C}^b(X; \mathbf{R})$, separating the points of X and containing the constant functions, be dense in $\mathscr{C}^b(X; \mathbf{R})$, it is necessary and sufficient that X be compact. (Let $\beta X$ be the Stone-Čech compactification of X (IX, p. 9). Prove that if $\beta X - X$ is not empty, there are non-dense subalgebras of $\mathscr{C}^b(X; \mathbf{R})$ separating the points of X and containing the constant functions, by observing that $\mathscr{C}^b(X; \mathbf{R})$ is identified with $\mathscr{C}(\beta X; \mathbf{R})$).