Let $(X, \mathcal{B}, \mu)$ be a Lebesgue probability space. Let $\Phi$ be the space of all invertible measure preserving transformations on $(X,\mathcal{B}, \mu )$, endowed with the weak topology. Given ergodic $T\in \Phi$, let $\Phi_T = \{ \phi^{-1} \circ T \circ \phi : \phi \in \Phi \}$ be the conjugacy class of $T$ with the subspace topology from $\Phi$. Let $C (T) = \{ \phi \in \Phi : \phi \circ T = T \circ \phi \}$ be the centralizer of $T$. For $\phi, \psi \in \Phi$, define an equivalence relation by $\phi \sim \psi$, if $\phi \circ \psi^{-1} \in C (T)$. Define the quotient space $Q (T) = \Phi / C (T)$. The map $h$ given by $h(\phi )=\phi^{-1}\circ T\circ \phi$ is a well-defined continuous bijection between $Q(T)$ and $\Phi_T$. Is $h$ a homeomorphism (i.e. $h^{-1}$ continuous)?
I believe the answer should be known, but I am unable to find a reference.