the relation between the sigma-algebras of two homeomorphic spaces

Question

It crosses my mind the following question : if $(X, \mathscr T)$ and $(Y, \mathscr S)$ are two homeomorphic spaces what can we say about the Borel sigma-algebras associated to each of them, otherwise what is the relation between $\sigma(\mathscr T)$ and $\sigma(\mathscr S)$ ?

Answer 1

Say you have a bijection $f: X \to Y$. If $f$ is continuous, it is measurable with respect to the Borel sigma-algebras, and hence the sigma-algebra on $Y$ pulls back to a subset of the sigma-algebra on $X$. If we identify $X$ and $Y$ via the bijection, we have $\sigma(Y) \subseteq \sigma(X)$. This shows that if $f$ is a homeomorphism (i.e. has a continuous inverse) $\sigma(X) = \sigma(Y)$. On the other hand, if $f$ is not continuous and does not have continuous inverse, there's not much you can say in general about the relationship of the two sigma-algebras.