$X=\{C[0,1],||_{\infty}\}$
$F=\{f\in X : f(1/2)=0\}$
$G=\{g\in X : g(1/2)\neq 0\}$
I need to find which one is open and which one is closed set in $X$.
Well $F$ is closed I guess, as if say $h(x)$ be a limit point of $F$ so there exist sequence $s_n(x)\in F$ such that $s_n(x)\rightarrow h(x)$ so $lim_{n\rightarrow\infty}s_n(x)=h(x)\forall x\in[0,1]$ so $h(1/2)=0$ so $h(x)\in F$ so $F$ is closed and $F^c=G$ is open.