TITLE: Point set topology question: compact Hausdorff topologies
QUESTION [5 upvotes]: $\tau_1,\tau_2,\tau_3$ are topologies on a set such that $\tau_1\subset \tau_2\subset \tau_3$ and $(X,\tau_2)$ is a compact Hausdorff space. Could any one tell me which of the following are correct?

$\tau_1=\tau_2$ if $(X,\tau_1)$ is compact Hausdorff.
$\tau_1=\tau_2$ if $(X,\tau_1)$ is compact.
$\tau_2=\tau_3$ if $(X,\tau_3)$ is Hausdorff.
$\tau_2=\tau_3$ if $(X,\tau_3)$ is compact.

REPLY [11 votes]: Hint: The identity mapping $(X,\tau_{i+1}) \to (X,\tau_i)$ is continuous and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism. This takes care of two statements and the two others are refuted by considering the trivial and the discrete topology on an infinite compact Hausdorff space $(X,\tau_2)$.