The author defines locally convex sets as a TVS whose topology is defined by a family of seminorms $\mathcal{P}$ such that $\bigcap_{p \in \mathcal{P}} \{x\ |\ p(x) = 0\} = \{0\}$. A bit later he gives the following example.
1. 5. Example Let $X$ be completely regular and let $C(X) =$ all continuous functions from $X$ to $\mathbb{F}$. If $K$ is a compact subset of $X$, define $p_K (f) = \sup \{|f(x)|\ |\ x \in K\}$. Then $\{p_K\ |\ \textrm{$K$ compact in $X$}\}$ is a family of seminorms that makes $C(X)$ into a LCS.
I don't see how the assumption of complete regularity is used in this example. If we want to prove that the above TVS is LCS, we just need to prove that for every nonzero function $f \in C(X)$ there exists some compact subset $K \subseteq X$ such that $p_K (f) \neq 0$. Since $f$ is nonzero, there has to exist a point $x_0 \in X$ such that $f(x_0) \neq 0$. But then we can just take $K = \{x_0\}$ because singletons are compact in any topology.
Clearly, I am missing something obvious, but I can't figure out what. Any help is greatly appreciated.