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 \(\displaystyle f(A)= A\)