Are homeomorphic, Hausdorff topologies on a set equal?

Question

Let $\mathcal T$ and $\mathcal S$ be two topologies on a set $X$ and $(X,\mathcal T)$ and $(X,\mathcal S)$ be homeomoric and compact and Hausdorff.

Is $\mathcal S$ equal to $\mathcal T$?

I know the answer is No (one can relabel the elements and get a homeomorphic one). But I'm looking for a good counterexample.

Answer 1

Let $X = \mathbb{N}$ and let

$\tau_1 = \mathcal{P}(\mathbb{N} \setminus \{1\}) \cup \{U\subseteq \mathbb{N}: 1\in U\textrm{ and } \mathbb{N}\setminus U \textrm{ is finite } \}$ and

$\tau_2 = \mathcal{P}(\mathbb{N} \setminus \{2\}) \cup \{U\subseteq \mathbb{N}: 2\in U\textrm{ and } \mathbb{N}\setminus U \textrm{ is finite } \}$.

(Note that $\mathcal{P}(.)$ denotes the power set.)

Then $(\mathbb{N},\tau_1)$ and $(\mathbb{N},\tau_2)$ are homeomorphic, and both topological spaces are compact and Hausdorff, but $\tau_1 \neq\tau_2$ because $\{2\} \in \tau_1 \setminus \tau_2$.