# 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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### Why do we have $E_0(\Bbb N^\Bbb N)\sim_c E_0$?

Let $E_0$ and $E_0(\Bbb N^\Bbb N)$ be the relations of eventual agreement on $2^\Bbb N$ and $\Bbb N^\Bbb N$, so $$xE_0y\iff\exists m\forall n\geq m x(n)=y(n)$$ and similarly for $E_0(\Bbb N^\Bbb N)$. ...
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### Baire space is zero dimensional

Here is the basis of Baire's topology with $s\in A^{<\mathbb{N}}$ and $\mathcal{B}=\{N_{s}:s\in A^{<\mathbb{N}}\}$ I was trying to show that elements of $\mathcal{B}$ is both open and closed. I ...
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### On gaps in the dimension of subspaces

Let $P\subseteq\Bbb N\cup\{\infty\}$, with $0\in P$. Does there exist a "nice" space $X_P$ such that $X_P$ has a subspace of dimension $n$ iff $n\in P$? By nice I mean nice enough that there'...
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### Uniformization property of $\mathbf{\Sigma}^0_n(\mathscr X\times\omega)$ for $n>1$

This question is about exercise 1C.6 from Moschovakis. The exercise is to prove if $P\subset\mathscr X\times \omega$ is in $\mathbf{\Sigma}^0_n$ for $n>1$, then there is some $P^*\subset P$ that is ...
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### Models for which Measurable implies Universally Baire?

To begin, Universally Baire sets are Lebesgue measurable (slide 4 of http://www.ub.edu/RSTR2018/slides/Woodin.pdf). I was wondering if there are any natural models of ZFC where the converse holds, and ...
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### On positive dimensional Polish spaces in which every compact set has empty interior

A standard characterization of the Baire space is that is the only nonempty, zero dimensional, Polish space in which every compact set has empty interior (up to homeomorphism of course). I'm ...
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### Every compact subset of Baire Space (ω^ω) has empty interior [duplicate]

Here is the Well known proposition. I assumed that K is the compact subset of ω^ω and it contains an open set Ns, so I thought if K is compact so Ns is because Ns is clopen set, If I can not find a ...
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### Every compact subset of Baire space $\omega^\omega$ has empty interior. [duplicate]

Here is the well-known proposition, I assumed that $K$ is compact subset of $\omega^\omega$ and it contains an open Set $N_s$, so $N_s$ is clopen and thus it is compact. If I can show $N_s$ can not be ...
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### Does the family of compact subsets of a Polish space belong to the Effros Borel structure?

Since the answer to my previous question (see Effros Borel structure ) turned out to be negative, I don't know how to solve the following problem: Let $X$ be a Polish space and let $\mathcal{F}(X)$ be ...
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### Effros Borel structure

Let $X$ be a Polish space and denote by $\mathcal{F}(X)$ the set of all closed subsets of $X$. The Effros Borel structure on $\mathcal{F}(X)$ is the smallest $\sigma$-algebra on $\mathcal{F}(X)$ ...
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In ZF we can prove Borel determinacy. In ZF we can prove that the determinacy for closed sets is equivalent to the Axiom of Choice. Since closed sets are Borel, it follows that the Axiom of Choice is ...
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### Homeomorphic images of endpoints of the Cantor set.

A Cantor set is a homeomorphic image of the standard ternary Cantor set $T$. Suppose that we have a Cantor set $C$ on the plane. It is well know that $C$ is in fact ambiently homeomorphic to $T$, ...
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### The supremum of uncountably many upper semmianalytic functions

I've been trying to learn a bit about analytic sets, and upper/lower semianalytic functions through the book, Stochastic optimal control, the discrete-time case. In chapter 7, Lemma 7.30, it said that ...
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### Iteratively constructing Borel sets

From this post, given a collection of sets $\mathcal{C}_0$, we can construct $\sigma(\mathcal{C}_0)$ through an iterative process, where for each step we let $\mathcal{C}_\alpha$ be the collection of ...