$\overline{F^c \cap K^c}$

Question

If given that $K$ is a compact subset and $F$ is closed subset then what is the nature of $\overline{F^c \cap K^c}$. I know that this is always closed hence if we consider any in $R^n$ with usual topology it will be compact but how to find a counter example or prove it in general.

Answer 1

Take (in $\Bbb R$, usual topology) $F=\{0\}, K=[0,1]$, which fits your demands.

Then $K^\complement \cap F^\complement = (-\infty,0) \cup (1,+\infty)$ with closure $(-\infty,0] \cup [1,+\infty)$, quite non-compact, for not bounded.