Define what it means for a topological space to be compact. Define what it means for a topological space to be Hausdorff.

Prove that a compact subspace of a Hausdorff space is closed. Hence prove that if $C_{1}$ and $C_{2}$ are compact subspaces of a Hausdorff space $X$ then $C_{1} \cap C_{2}$ is compact.

A subset $U$ of $\mathbb{R}$ is open in the cocountable topology if $U$ is empty or its complement in $\mathbb{R}$ is countable. Is $\mathbb{R}$ Hausdorff in the cocountable topology? Which subsets of $\mathbb{R}$ are compact in the cocountable topology?