# 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.

607 questions
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### Combinations of Lebesgue measurability, the property of Baire and the perfect set property

Lebesgue measurability (LM), the property of Baire (BP) and the perfect set property (PSP) are probably the most prominent among all the regularity properties of sets of reals. Such a set can either ...
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### Why use transfinite induction？

Why use transfinite induction to prove? I think the inclusion relation is trivial by transfinite recursion. $\mathbf{11.B}$ The Borel Hierarchy Assume now that $X$ is metrizable, so that every ...
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### Analytic sets are Lebesgue measurable

Analytic subsets of $\mathbb{R}$ are projections of Borel sets in $\mathbb{R}^2$. I'm trying to understand a proof that these sets are always Lebesgue measurable. One can first prove that analytic ...
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### Vector space of real valued function with size continuum

I know how I can construct a function such that $f^{-1} (y)$ has size less than continuum actually countable. Here is the proof, Define a relationship as following x\sim y \ \text{iff} \ x-y\...
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### A continuum-sized convenient category of topological spaces

From the concluding section of Quasi-Polish Spaces by Matthew de Brecht: "It turns out that the category of quasi-Polish spaces and continuous functions has a very natural description: up to ...
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### Almost clopen sets and Baire property?

A set $A$ is said to have the Baire property if it differs from an open set by a meagre set, that is, if there exists an open set $G$ such that $A\Delta G$ is meager (where $\Delta$ denotes the ...
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### Given the hierarchy of Borel sets, how to prove that ${\bf \Sigma}_\alpha^0 \subsetneq {\bf \Sigma}_{\alpha+1}^0$ for all ordinal $\alpha<\omega_1$

My textbook Introduction to Set Theory 3rd by Hrbacek and Jech introduces $\mathcal B$, the set of all Borel sets, and then mentions that I have tried, but only to successfully prove that there ...
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### What uniquely characterizes equivalence classes of eventually equal binary sequences?

Let $X$ be the set of all infinite binary sequences. (Or we can think of them as subsets of $\mathbb{N}$ or real numbers between $0$ and $1$.) Let us define an equivalence relation $\sim$ on $X$ by ...
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### When does a subset of a Polish space meet all the orbits?

While I was studying Borel actions of Polish groups on Polish spaces (I assume of measure $1$), I have tried to understand if a measure $1$ (hence dense) subset of this Polish space meets all the ...
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### How to Modify a Borel function in a Borel way-Self study

I have the following questions: assume that $X$ be a standard Borel space (i.e. a Polish space equipped with the $\sigma$-algebra generated by open sets) with a (possibly Borel, invariant, ergodic) ...
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### Almost clopen sets

A set $A$ is said to be almost clopen if it can be uniquely represented in the form $A=B\Delta M$, where $B$ is a clopen set, $M$ is a set of first category (or meagre), and $\Delta$ denotes the ...
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### Is every $\sigma$-algebra a Borel $\sigma$-algebra?

Let $\mathscr B$ be a $\sigma$-algebra on a set $X$. Does there always exist a topology $\tau$ on $X$ such that $\mathscr B$ is the Borel $\sigma$-algebra with respect to $\tau$, that is, $\mathscr B$ ...
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### Player II has a winning strategy in the *-game, G*(A), iff A is countable

I have a question concerning the proof of this theorem in Kechris' "Classical Descriptive Set Theory." The rules of the game i as follows: Let $X$ be a nonempty perfect Polish space with compatible ...
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### Is there a dense set in $\mathbb{R}$ with outer measure $1$? And what about with the Density Topology?

I'm looking for a dense set in $\mathbb{R}$ with outer measure $1$. There is an example like that, but in $[0,1]$ Vitali set of outer-measure exactly $1$.. Also I've tried with Bernstein sets but I ...
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### Is the projection of a Jordan measurable set Jordan measurable?

The projection of a Borel set need not be a Borel set, and the projection of a Lebesgue measurable set need not be Lebesgue measurable. My question is, is the projection of a Jordan measurable set ...
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### Lebesgue Measurable Set which is not a union of a Borel set and a subset of a null meager set?

This is a follow-up to my question here. The Lebesgue Sigma algebra is the completion of the Borel Sigma algebra under the Lebesgue measure, which means that every Lebesgue measurable set can be ...
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### Is every Borel subset of a measurable set measurable?

Let $A$ be a Lebesgue measurable subset of $\mathbb{R}$. Let us consider the subspace topology on $A$, and let us consider the Borel sigma algebra under that topology. My question is, is every Borel ...
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### A dense measure $0$ $G_\delta$ subset of the Fat Cantor set?

The fat Cantor set is a nowhere dense subset of $\mathbb{R}$ with positive Lebesgue measure. My question is, does there exist a $G_\delta$ set dense in the fat Cantor set with Lebesgue measure $0$? ...
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### A perfect nowhere dense set which intersects every open set with positive measure?

A perfect set is a closed set with no isolated points. A nowhere dense set is a set whose closure has empty interior. My question is, what is an example of a nonempty perfect nowhere dense subset of ...
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### Why are there continuum many nowhere dense subsets of $\mathbb R$?

I am able to see why the closed nowhere dense subsets of $\mathbb R$ are equinumerous with $\mathbb R$: Every closed nowhere dense subset of $\mathbb R$ is the boundary of an open set (namely its ...
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### Lebesgue Measurable Set which is not a union of a Borel set and a subset of a null $F_\sigma$ set?

The Lebesgue Sigma algebra is the completion of the Borel Sigma algebra under the Lebesgue measure, which means that every Lebesgue measurable set can be written as a union of a Borel set and a subset ...
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### What is the Sigma Algebra generated by Jordan measurable sets?

Unlike Lebesgue measurable sets, Jordan measurable sets do not form a Sigma algebra. So my question is, what is the Sigma algebra $J$ generated by Jordan measurable sets? All intervals are Jordan ...
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### Example of a non-Borel set on Euclidean space

Can someone give an example of a subset of $\mathbb{R}^n$, $n >1$, that is not a Borel set?
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### Example of a Lebesgue measurable set that can't be constructed from Borel sets and projections?

The Borel sigma algebra on $\mathbb{R}^n$ is obtained by starting with open sets and repeatedly applying the operations of complement, countable union, countable intersection. Now Henri Lebesgue ...
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### Why doesn't this construction make the rational numbers a $G_\delta$ set?

A $G_\delta$ set is a countable intersection of open sets. The set of rational numbers is not a $G_\delta$ set. But I'm wondering why the following construction doesn't make it a $G_\delta$ set. ...
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### The Baire space as an automorphism group?

Wikipedia claims that the Baire space "is [the] automorphism group of [a] countably infinite saturated model $\mathfrak{M}$ of some complete theory $T$", however I see neither an obvious group ...
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### Are “most” sets in $\mathbb R$ neither open nor closed?

It seems intuitive to believe that most subsets of $\mathbb R$ are neither open nor closed. For instance, if we consider the collection of all (open, closed, half-closed/open) intervals, then one ...
### Can an uncountable subset of $\mathbb{R}$ have empty intersection with its derived set? [duplicate]
Question: Is there an uncountable subset $A$ of $\mathbb{R}$ such that $A\cap A'=\emptyset$, where $A'$ denotes the derived set of $A$? I just know that an uncountable set $A$ must have limit points ...