Weaker/Stronger Topologies and Compact/Hausdorff Spaces In my topology lecture notes, I have written:
"By considering the identity map between different spaces with the same underlying set, it follows that for a compact, Hausdorff space:
$\bullet$ any weaker topology is compact, but not Hausdorff
$\bullet$ any stronger topology is Hausdorff, but not compact "
However, I'm struggling to see why this is. Can anyone shed some light on this?

Hint. $X$ being a set, a topology $\tau$ is weaker than a topology $\sigma$ on $X$ if and only if the application
$$ (X, \sigma) \to (X,\tau), x \mapsto x $$
is continuous.
