# 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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### Is the following set Borel? [closed]

Define the set $$A = \{ x \in \mathbb{R} \, \vert \, \forall M \in \mathbb{R}: \exists p, q \in \mathbb{Z}: p> M \text{ and }0 < \vert x-\frac{q}{p} \vert < \frac{1}{p^{3}} \}.$$ Is this ...
1 vote
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### Ergodic transformations form a $G_\delta$ set in the weak topology of the automorphism group.

If $(X,\mathcal{L},\mu)$ is a Lebesgue-standar space and $G$ is its group of automorphisms, i know that the set of all ergodic transformations $\mathcal{E}$ is a $G_\delta$ set in the weak topology. ...
1 vote
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### Show that the following metric is complete on $L_1(X,Y)$

I am working through the exercises of Kechris' Classical Descriptive Set Theory for personal edification. I've solved Exercise 3.5, but the proof was two pages. Did I miss a more straightforward ...
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### Are projective sets from descriptive set theory projective objects in Set category?

There are projective sets in descriptive set theory. For them, the axiom of determinacy is not contradicting the axiom of choice. Given the axiom of choice, every set is a projective object. But in a ...
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### Why isn't $[0,1]^{\aleph_1}$ isomorphic to $[0,1]$ as a measure space.

I was talking with a professor, and he mentioned that $[0,1]$ (with Lebesgue $\sigma$-algebra and Lebesgue measure) isn't isomorphic as a measure space to $[0,1]^{\aleph_1}$ (with the product measure)....
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### Alpha recursion - Constructible universe and Analytical hierarchy

Alpha recursion and Constructible universe are very intertwined, because the first is based on the concept of admissible ordinal $\alpha$ which is defined as an ordinal such that $L_\alpha$ - a set ...
1 vote
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### Two set-theoretical assumptions

My question about the following two set-theoretical assumptions: union of less than continuum many meager subsets of $\Bbb R$ is meager in $\Bbb R.$ Union of less than continuum many meager subsets ...
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### Locus of Continuity of a Function between Polish Spaces

I think the question is maybe a bit trivial, but right now my head doesn't work that much well so I'm asking y'all to have some clarification. I know that, for a subset $A\subseteq \mathbb R$, the ...
1 vote
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### Countable dense subset of Cantor space isn't Wadge reducible to its complement

I am trying to prove that a countable dense subset $Q\subset 2^\omega$ (where $2^\omega$ denotes the Cantor space) is not Wadge reducible to its complement. I am trying to prove this because I want to ...
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### Hyperreals and Suslin problem

Hyperreal numbers $^*\mathbb{R}$ form a totally ordered field that is a proper extension of the real numbers (i.e. non-isomorphic to real numbers $\mathbb{R}$). It is obvious that $^*\mathbb{R}$ has ...
1 vote
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I am trying to solve Ex 20.10 on Kechris descriptive set theory book. Throughout this post, a "game on $A$" refers to the infinite game $G(T,X)$ where $T$ is a pruned tree $T\subset A^{<\... 0 votes 1 answer 51 views ### Descriptive Set Theory - definable sets can be partitioned into more pieces than elements of original set [closed] I just read that "In Descriptive Set Theory .... definable sets can .... counter intuitively .... be partitioned into more pieces than there are elements in the original set". Would anyone ... 1 vote 1 answer 70 views ### If a metric space$F$is Borel isomorphic to a Lusin measurable space, then$F$must be homeomorphic to a Borel subset of some compact metric space I'm reading about Lusin and Suslin spaces from page 46 of Dellacherie/Meyer's Probabilities and Potential. Here I restrict myself to metric spaces. A Borel isomorphism between two measurable spaces ... 3 votes 0 answers 71 views ### How to prove these definitions of an analytic set are equivalent? Let$F$be a set and$\mathcal F$a collection of subsets of$F$such that$\emptyset \in \mathcal F$. We denote by$F_\sigma$(resp.$F_\delta$) the closure of$F$under countable union (resp. ... 5 votes 1 answer 128 views ### Question on Gale-Stewart Theorem and Axiom of Choice I am reading Kechris' descriptive set theory text book, and there is this Theorem regarding infinite games: Gale-Stewart: Let$T$be a non-empty pruned tree on$A$. Let$X\subset[T]$be closed or open.... 1 vote 0 answers 138 views ### Proving that the set of points where a continuous function is differentiable is a$\mathbf{\Pi_3^0}$set Let$f:\mathbb{R}\to\mathbb{R}$be a continuous function and let$D$be the set of points where$f$is differentiable. In order to show that$D\in\mathbf{\Pi_3^0}(\mathbb{R})$, I want to show that$x\...
Question: Show that if $0 \leq \text{m}(A) \leq \infty$, then for each positive $q < \text{m}(A)$ there is a perfect set $B \subset A$ of measure $q$. (All subsets are in $\mathbb{R}$ and the ...
On Wikipedia there's a nice example of a non-Borel set due to Luzin. For completeness, I'll summarize it here. For $x\in[0,1]$, let \begin{align} x=a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 +...