I'm working on Exercise 2.28 in Prof. David Marker's notes http://homepages.math.uic.edu/~marker/math512/dst.pdf on refining the topology to make Borel sets clopen.
Question: Suppose $X$ is a Polish space and $ B \subseteq X \times X $ is Borel. Is it always possible to put a new Polish topology on $X$ such that $B$ is clopen in the new product topology on $X \times X$?
Theorem 2.24 in the notes above tells us that we can find a new topology $\tau'$ on $X \times X$ such that $B$ is clopen, then is it possible to make this $\tau'$ a product topology?
What I can see is, maybe we can refine $\tau'$ again with projection maps on each cordinate. Denote it as $\tau''$. But it only shows that $\tau''$ is at least finer than the product topology.
Any help will be appreciated!
Nice question! Here's a hint to start you off: is it possible to have an uncountable separable metric space $X$ such that the diagonal $\Delta:=\{(x,x) : x \in X\}$ is an open subset of the product space $X \times X$?