Every weak closed set is convex?

Question

Let $X$ a locally convex topological vector space and $C\subseteq X$ a weak closed set. Is $C$ convex? I think it must be, because the weak closure of $C$ is the intersection of all weak closed semi-spaces that contain $C$, and all of them are convex.

Answer 1

No. Consider the real line $\mathbb{R}$ and the set $C = \{-1,1\}$. $C$ is (weakly) closed but not convex.

Answer 2

Remember, the closed convex hull is the intersection of (necessarily weakly) closed half-spaces containing $C$. The weak closure is the intersection of all weakly closed sets containing $C$. For example, you can union two disjoint (parallel) closed half-spaces together. The complement is the intersection of two open half-spaces, and hence is weakly open.

Answer 3

"because the weak closure of C is the intersection of all weak closed semi-spaces that contain " the prove of it coontains the Hahn-Banach theorem if C is convex. So your reason is not true.