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.

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.

883 questions
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Cardinality of Borel sigma algebra

It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, and actually hard to find a proof of it. Can…
60
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The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$

I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$. Let me try to elucidate my understanding of the…
Harry Williams
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What is the cardinality of the set of all topologies on $\mathbb{R}$?

This was asked on Quora. I thought about it a little bit but didn't make much progress beyond some obvious upper and lower bounds. The answer probably depends on AC and perhaps also GCH or other axioms. A quick search also failed to provide answers.
Alon Amit
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Why do we study Polish spaces?

In descriptive set theory, a lot of space is devoted to properties of Polish spaces. (A Polish space is a topological space, which is separable and completely metrizable.) I would like to know why there is so much interest in this class of spaces.…
Martin Sleziak
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41
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Examples of uncountable sets with zero Lebesgue measure

I would like examples of uncountable subsets of $\mathbb{R}$ that have zero Lebesgue measure and are not obtained by the Cantor set. Thanks.
40
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Is there a $\sigma$-algebra on $\mathbb{R}$ strictly between the Borel and Lebesgue algebras?

So, after proving that $\mathfrak{B}(\mathbb{R})\subset \mathfrak{L}(\mathbb{R})$, I asked myself, and now asking you, is there a set $\mathfrak{S}(\mathbb{R})$, which satisfies: $$\mathfrak{B}(\mathbb{R} )\subset \mathfrak{S}(\mathbb{R})\subset…
Salech Alhasov
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Where do Cantor sets naturally occur?

Cantor sets in general of course have many interesting properties on their own, and are also often used as examples of sets with these properties, but do they naturally occur in any application?
malin
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36
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Correspondences between Borel algebras and topological spaces

Though tangentially related to another post on MathOverflow (here), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but... Suppose $\Sigma$ is a sigma-algebra over a set, $X$. For any…
32
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Continuous functions are differentiable on a measurable set?

I came across the following challenging problem in my self-study: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Then the set of points where $f$ is differentiable is a measurable set. I am having trouble thinking of where to begin in…
Vulcan
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31
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What's the difference between rationals and irrationals - topologically?

I know that sets of rational and irrational numbers are quite different. In measure, almost no real number is rational and of course, $\mathrm{card}(\mathbb Q) < \mathrm{card}(\mathbb R \setminus \mathbb Q) $ tells us that there are indeed much…
Dario
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28
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Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly

While reading David Williams's "Probability with Martingales", the following statement caught my fancy: Every subset of $\mathbb{R}$ which we meet in everyday use is an element of Borel $\sigma$-algebra $\mathcal{B}$; and indeed it is difficult…
25
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1 answer

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$? Recall the definition: Let $S = \mathbb{N^{
25
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1 answer

Universally measurable sets of $\mathbb{R}^2$

$$\text{Is }{{\cal B}(\mathbb{R}^2})^u={{\cal B}(\mathbb{R}})^u\times {{\cal B}(\mathbb{R}})^u\,?\tag1$$ Is the $\sigma$-algebra of universally measurable sets on $\mathbb{R}^2$ equal to the product $\sigma$-algebra of two copies of the universally…
23
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3 answers

Countable compact spaces as ordinals

I heard at some point (without seeing a proof) that every countable, compact space $X$ is homeomorphic to a countable successor ordinal with the usual order topology. Is this true? Perhaps someone can offer a sketch of the proof or suggest a…
19
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1 answer

Is there a set $A \subset [0,1]$ such that both $A$ and $[0,1] \setminus A$ intersect every positive-measure set?

Is there a set $A \subset [0,1]$ such that for every Borel set $B \subset [0,1]$ of positive Lebesgue measure, both $B \cap A$ and $B \setminus A$ are non-empty? This is, in a sense, the "measure-theoretic analogue" of the more obvious topological…
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