i'm studying this Theorem 3.4, Rudin's Functional Anlysis, page 59, buy i'm a little bit stuck in some parts of the proof.
The Theorem says: Suppose $A$ and $B$ are disjoint, non empty, convex sets in a topological vector space $X$.
a) if $A$ is open there exist $\Lambda \in X^{*}$ and $\gamma \in \mathbb{R}$ such that $$\Re (\Lambda x) < \gamma \leq \Re (\Lambda y),$$ for every $x\in A$ and for every $y \in B$.
But i don't understand how to use that $A$ is an open set, i mean in the proof says that $f(A)$ is open because $f$ is an open map, but i don't get at all why this is too important, i will really appreciate any hint.