Exercise 6, § 4

Let $X$ be a completely regular space and let $C^\infty(X; \mathbf{R})$ denote the normed subalgebra of $B(X; \mathbf{R})$ consisting of all bounded continuous real-valued functions on $X$. In order that every subalgebra of $C^\infty(X; \mathbf{R})$, which separates the points of $X$ and contains the constant functions, should be dense in $C^\infty(X; \mathbf{R})$, it is necessary and sufficient that $X$ should be compact. [Let $\beta X$ be the Stone-Čech compactification of $X$ (Chapter IX, § 1, Exercise 7). Show that if $\beta X - X$ is not empty then there are non-dense subalgebras of $C^\infty(X; \mathbf{R})$ which separate the points of $X$ and contain the constant functions, by observing that $C^\infty(X; \mathbf{R})$ may be identified with $C(\beta X; \mathbf{R})$.]