An **isomorphism** between two measurable spaces $(X\_1,\mathcal{B}\_1), (X\_2,\mathcal{B}\_2)$ is a measurable bijection $f:X\_1\rightarrow X\_2$ whose inverse is also measurable.


**QUESTION.** Can there be an **isomorphism** between an uncountable Polish space and a non-Hausdorff topological space, each endowed with its respective Borel $\sigma$-algebra?



Polish spaces and isomorphisms
Here is a particularly simple way to see why the answer is yes, as YCor already pointed out in a comment. Uncountability is a red herring.


Ignore the uncountability condition for now. Consider the set $\{0,1\}$ once with the discrete topology and once with the non-Hausdorff topology whose only nontrivial open set is $\{1\}$, the [Sierpiński space](https://en.wikipedia.org/wiki/Sierpi%C5%84ski_space). They are clearly Borel isomorphic.


Now take your favorite uncountable Polish space $X$. Taking the disjoint union $X\sqcup\{0,1\}$ of both two point spaces will again give you two Borel isomorphic spaces, only one of which is Polish.


