Recall that a strong Choquet space is one where player II has a winning strategy in the game where two players take turns: player I chooses an open set and a point inside, then player II chooses a smaller open set containing the chosen point, then player I chooses a smaller still open set and a point inside, and so on, and player II wins if the intersection is nonempty after $\omega$ turns.

Suppose we have two compact Hausdorff spaces $X,Y$ and a continuous surjection $f\colon X\to Y$. Suppose in addition that we have $Y'\subseteq Y$ such that $f^{-1}[Y']$ is strong Choquet. Is it true, then, that $Y'$ is strong Choquet as well?

If $X,Y$ are metrisable/second countable, then this is true: strong Choquet subspaces are exactly the $G_\delta$ subsets, and it follows from e.g. exercise 24.20 in Kechris that if $f^{-1}[Y']$ is $G_\delta$, then so is $Y'$. But what about in general?

Also, if $f$ was open, this would be true (as continuous open images of strong Choquet spaces are strong Choquet), but $f$ is in general closed and not open.

If it helps, we may assume that $X,Y$ are both zero-dimensional, i.e. have bases of clopen sets.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.