I have a lema and an example for which i don't understand how are they not in contradiction. I am not sure is there something obvius that i am missing. Any help would be much appreciated. Here it is:
Lema: Let $V$ be a real vector space and $S\subseteq V$ be a convex cone. If $F$ is a proper face of $F$ then $S\setminus F$ is dense in $S$ with respect to any vector topology on $V$.
Example: Consider convex cone $S=\{f\in \mathcal{S}(\mathbb{R}^{n}):f>0\}$ in Schwartz space $\mathcal{S}(\mathbb{R}^{n})$ of rapidly decreasing real-valued functions on $\mathbb{R}^{n}$. For every element $f\in S$, the face $F_S(f)=(\mathbb{R}^{+}f-S)\cap S$ (this is smallest face containing $f$) is a dense proper face of $S$.
My question is, how can be that $F$ is dense in $S$ and in the same time $S\setminus F$ is dense in $S$?