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Since every Polish space is Borel isomorphic to a zero-dimensional compact metric space, it suffices to characterize topological spaces, which are Borel isomorphic to a zero-dimensional compact metri …

This is a very interesting question whose answer depends on dimension properties of the spaces $X,Y$. First we introduce a suitable terminology. A function $f:X\to Y$ between topological spaces is cal …

I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone. Theorem. The linear hull of any linearly independent Borel set in a Polish topolo …

Yes, this is true: by Exercise 13.5 in Kechris' "Classical Descriptive Set Theory", for any Borel function $f:X\to Y$ between Polish spaces there exists a continuous bijective map $i:Z\to X$ from a Po …

Problem 1. Is it true that for every cardinal $\kappa\le\mathfrak c$ there exists a partition $(B_\alpha)_{\alpha\in\kappa}$ of the real line into $\kappa$ pairwise disjoint non-empty Borel subsets? R …

The answer to this question is negative and follows from Theorem. Each $G_{\delta\sigma}$-subset $A$ of a Polish space $X$ can be written as the union $\bigcup_{n\in\omega}A_n$ of a sequence $(A_n)_{n …

Problem. Assume that a metrizable separable space $X$ is the countable union $X=\bigcup_{n\in\omega}X_n$ of pairwise disjoint $G_\delta$-sets $X_n$ in $X$ such that each $X_n$ is an absolute $F_{\sig …

Just a small addition to the existing answers. Theorem. There exists a non-metrizable compact Hausdorff space $K$ admitting a continuous surjective function $f:K\to[0,1]^2$ to the unit square such …

If a map $f:Y\to X$ is an open surjection, then the inverse map $f^{-1}:X\to \mathcal K(Y)$ to the hyperspace $\mathcal K(Y)$ is continuous. The hyperpsace $\mathcal K(Y)$ is the space of nonpempty co …