The sets $\{\{f\in C(X):|g−f|\leq u\} : g\in C(X) \text{ and } u \text{ a positive unit of } C(X)\}$ form a base for some topology on $C(X)$ which is called the $m$-topology on $C(X)$. A norm on $C^{*}(X)$ is given by $\|f\|=\sup|f(x)|$ and the resulting metric topology is called the uniform norm topology on $C^{*}(X)$.
How to show the above two topologies on $C^{*}(X)$ (consider relative top. for the m-topology) coincide iff $X$ is pseudocompact.
(Hints: When $X$ is not pseudo-compact, the set of constant functions in $C^{*}(X)$ is discrete, in the $m$-topology, so that $C^{*}(X)$ is not even a topological vector space whereas $C^{*}(X)$ forms a Banach algebra w.r.t. uniform norm topology).