I try to figure out under which conditions the Graph of a continuous surjection $\pi:X\to Y$ (think quotient map) between topological spaces $X$ and $Y$ is measurable with respect to the product-$\sigma$-algebra.
I could reduce it to the measurability of the diagonal $$ \Delta_Y = \{ (y,y) | y \in Y\} $$ with respect to $B \otimes B$. Where $B$ is the Borel-$\sigma$-algebra on $Y$.
So my question is: Under which topological condition is the diagonal measurable in the product of the Borel-$\sigma$-algebra. In particular I want to know conditions on compact Hausdorff spaces.
The situation is clear for metrizable compact spaces as then the product of the Borel-$\sigma$-algebra is the Borel-$\sigma$-algebra of the product (and the diagonal is closed by Hausdorffness). I would like to see any condition that is weaker than second countability (i.e. metrizability). Or is this in fact equivalent to metrizability?
I already found in a paper of Hoffmann-Jørgensen ("Existence of Conditional Probability Measures", Theorem 3) that the measurability of the diagonal is equivalent to the existence of a countably generated sub-$\sigma$-algebra of the Borel-$\sigma$-algebra on $Y$ which contains the points. This is in turn equivalent to the existence of an measurable injection $f:Y \to [0,1]$.
So yet again I can reframe my question to: What topological properties ensure the existence of a Borel-measurable injection to the compact interval?
Again this is clearly true for Polish spaces but I hope for a non-metrizable class of spaces with that property.
Thank you all!