# Questions tagged [descriptive-set-theory]

In descriptive set theory we mostly study Polish spaces such as the Baire space, the Cantor space, and the reals. Questions about the Borel hierarchy, the projective hierarchy, Polish spaces, infinite games and determinacy related topics, all fit into this category very well.

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### Total disconnection and zero-dimension in Polish spaces

First of all Polish spaces are completely-metrizable, separable topological space and by zero-dimensional Polish space I mean that the Polish space has a (countable) basis made of clopen sets. It is ...
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Given that $X$ and $Y$ are Polish spaces and $E$ and $F$ are Borel equivalence relations on $X$ and $Y$, respectively, we say that $E$ is Borel reducible to $F$ if there exists a Borel function $f:X\... 1answer 122 views ### Subsets of$\mathcal{P}_{\infty}\mathbb{N}$that are open and dense for the Ellentuck topology are completely Ramsey I am reading Chapter 10 of Albiac and Kalton's book$\textit{Topics in Banach Space Theory}$, and am trying to understand the proof of Theorem 10.1.3, namely that subsets of$\mathcal{P}_{\infty}\...
Let $A\subseteq X\times Y$, with $X$ and $Y$ Polish spaces. Suppose that $A=\bigcup_{c\in C}A_{c}$, where $C\subseteq X$ is a closed set and each $A_{c}$ is Borel. Can we conclude that $A$ is ...
The language of the first-order theory of real closed fields consists of the non-logical symbols $0$, $1$, $+$, $\cdot$, $<$, and $=$. My question is, for what subsets $X$ of $\mathbb{R}$ does ...