Normed space and convex hull of closed subset

Question

Let $(V, ||\cdot||)$ be a normed space. If $ C\subseteq V$ is a closed set we do not know if $ch(C)$ is closed or not.

The professor provided this example that as of now I'm not getting: Consider the set $$S=\big(\{0\}\times[0,1]\big) \cup \big([0,+\infty]\times\{0\}\big) \subseteq \mathbb R^2.$$ As the union of two closed sets, $S$ is closed. However $$ch(S)=\big([0,+\infty)\times[0,1)\big)\cup \{(0,1)\}$$ which is not closed. I don't get how $ch(S)$ can not be closed.

Answer 1

Observe that $$ \{1\}\times [0,\infty)\subset \overline{ch(S)}, $$ while $$ \{1\}\times [0,\infty)\cap {ch(S)}=\varnothing. $$