proof that the convex hull of $S$ is open if $S$ is open

Let $S$ be an open set in some topological vector space $V$. For any sequence of positive real numbers $\mathrm{\Lambda }=({\lambda }_1,\mathrm{\dots },{\lambda }_n)$ with ${\sum }_{i=1}^n{\lambda }_i=1$ define

$$S_{\mathrm{\Lambda }}=\{x\in V\text{ such that }x=\sum _{i=1}^n{\lambda }_i{s}_i\text{ for }{s}_i\in S\}.$$

Then since addition and scalar multiplication are both open maps, each $S_{\mathrm{\Lambda }}$ is open. Finally, the convex hull is clearly just

$$\bigcup _{\mathrm{\Lambda }}{S}_{\mathrm{\Lambda }},$$

which is therefore open.