homeomorphism of same space

homeomorphism of same space

If $(X,\mathcal T)$ and $(X,\bar{\mathcal T})$ are homeoporphic, isn't
that $\mathcal T=\bar{\mathcal T}$?
I ask this question because of a theorem: If $(X,\mathcal T)$ be compact
and $(X,\bar{\mathcal T})$ be Hausdorff, also given that $\bar{\mathcal
T}\subseteq\mathcal T$, then $\mathcal T=\bar{\mathcal T}$.
Proof: consider the function $f:(X,\mathcal T)\to(X,\bar{\mathcal T})$
defined by $f(x)=x$. Then it is bijection. It is also continuous because
$\bar{\mathcal T}\subseteq\mathcal T$. By the property of functions from
compact space to Hausdorff space, the function is a homeomorphism. So,
$\mathcal T=\bar{\mathcal T}$.
I don't understand the last step that why homeomorphic means same topology.