Prove that if A & B are disjoint open sets then their intersection is compact.

Question

Prove that if A & B are disjoint open sets then their intersection is compact. could anyone give me a hint on how to do this please?

My trial their intersection will be the empty set which is closed, but I do not know how to justify that it is bounded even though I am convinced that it is bounded. could anyone help me in this please? is my trial correct?

Answer 1

$A$ and $B$ being disjoint means that $A \cap B = \emptyset$.

$\emptyset$ is trivially compact since the collection $\emptyset$ (which is a finite subset of any cover of $\emptyset$) covers it.

This has nothing to do with $A$ and $B$ being open.