Prove

1) $\displaystyle X $ is a regular space, $\displaystyle A $ compact set and $\displaystyle B$ closed set,

$\displaystyle A \bigcap B = {\O} $

Then there is open sets U and V exist, such that

$\displaystyle A\subseteq U, \ B \subseteq V,\ A \bigcap B = {\O} $

2) $\displaystyle X $ is compact and hausdorff space and $\displaystyle f : X \rightarrow X $ is continuous function

prove there is a closed set non empty such that $\displaystyle f(A)= A$