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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.

970 questions
17
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Bijection between closed uncountable subset of $\Bbb R$ and $\Bbb R$.

This is probably a really stupid question but I'm hoping it's true: can we find a bijection between any uncountable closed subset of $\Bbb R$ and $\Bbb R$ (real number line)?
user140506
  • 319
12
votes
2 answers

Does every Lebesgue measurable set have the Baire property?

Title says it all, so I'll just repeat it: Does every Lebesgue measurable set have the Baire property?
Anonymous
  • 123
12
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2 answers

Borel hierarchy doesn't "collapse" before $\omega_1$

I know that $\bigcup_{\alpha < \omega_1} \Sigma_\alpha^0 = \mathcal{B}(X)$ for any Polish space $X$, but I want to know if it's really necessary to take the union for every ordinal less than $\omega_1$. Is there an example of a Polish space where…
Oliver Miller
  • 191
8
votes
1 answer

Borel Hierarchies

I have started reading about Borel Hierarchies. If I understand correctly, Then $\Sigma_2^0$ is the collection of all sets of the form $A=\bigcup_{n \in \omega}{B_n}^c$, where $B_n$ is open. also, $\Sigma_3^0$ is the collection of all sets of the…
topsi
  • 4,222
8
votes
1 answer

Every perfect subset (in $\mathbb{R}$) has cardinality $\mathfrak{c}$? (in ZF)

I have been reading "Set theory" of T. Jech. I saw a proof of "Every perfect subset (in $\mathbb{R}$) has cardinality $\mathfrak{c}$". Here is this proof: Proof: Given a perfect set $P$, we want to find a one-to-one function $F$ from…
Hanul Jeon
  • 27,376
8
votes
3 answers

Is it possible to divide the real line into two disjoint totally disconnected spaces of equal cardinality?

Is it explicitly possible to take $ \mathbb R$ and to divide it into two sets (say $\mathbb A$ and $\mathbb B$), which are : disjoint ($\mathbb A \cap \mathbb B = \emptyset$), totally disconnected (i.e. contain no open intervals) (EDIT:) and for all…
user68475
8
votes
2 answers

Complexity of the set of surjective continuous functions

Let $X,Y$ be complete separable metric spaces, with $X$ locally compact, and $C(X,Y)$ the space of continuous functions from $X$ to $Y$, equipped with the topology of uniform convergence on compact sets. If I am not mistaken, $C(X,Y)$ is…
Nate Eldredge
  • 97,710
7
votes
1 answer

Any two uncountable Borel subsets of $[0,1]$ Borel isomorphic

Question 1: Why are any two uncountable Borel subsets of $[0,1]$ Borel isomorphic? Recall that $f\colon X \to Y$ is a Borel isomorphism if $A$ being Borel in $X$ is equivalent to $f(A)$ being Borel in $Y$. In Kechris' Classical Descriptive Set…
Quinn Culver
  • 4,471
6
votes
2 answers

R subset and Borel set

Which is the simplest example of a subset of R that is not a Borel set? Ps: I asked my probability professor how I can see if a subset of R is a Borel set or not? He answered that in the case we will use the set are always Borel set and in fact he…
user43158
5
votes
1 answer

How to show $\Sigma_{\omega}^{0}$ (not ) exhaust the Borel sets?

Let$(X,\mathscr{T})$ be a topological space. The collection of Borel sets in $X$ is the smallest $\sigma$-algebra containing the open sets in $\mathscr{T}$. $\Sigma_{\omega}^{0}$ is inductively defined by the following collection of subsets of $X$.…
Metta World Peace
  • 6,403
5
votes
1 answer

Borel image of a Polish space

Let $X, Y$ be Polish spaces and $f : Y \to X$ be a Borel function, i.e. preimage of every Borel subset of $X$ be a Borel subset of $Y$. Prove that $f[Y]$ is an analytic subset of $X$. Note: If $f$ were continous, the above would be a precise…
Adayah
  • 10,468
4
votes
1 answer

The set of all points $x$ in which $f$ is continuous is $G_{\delta}$

in exercise 4.16 here: Jech - Set Theory, we are asked to prove that: Given a function $f: \mathbb{R} \rightarrow \mathbb{R}$, the set of all points $x$ in which $f$ is continuous is $G_{\delta}$. Something bothers me here since it seems that more…
topsi
  • 4,222
4
votes
1 answer

Where does the collection of sets with positive upper density lie in the Borel hierarchy?

The upper density of a set $A\subset\mathbb{N}$ is defined as $\bar{d}(A) = \limsup_{N\to\infty}\frac{|A\cap\{1,\ldots,N\}|}{N}$. If we identify a set $A$ with its characteristic function $\chi_A$, which we treat as an element of $2^{\mathbb{N}}$,…
tc1729
  • 3,017
3
votes
1 answer

Closure of $\mathbf{\Sigma_{n}^{0}} $ under finite cartesian products

An exercise in Moschovakis' descriptive set theory, asks to show if we have pointsets $P\subset\mathcal{X}$ and $Q\subset\mathcal{Y}$ both of which are $\mathbf{\Sigma_{n}^{0}}$, then to show that $P\times Q\subset\mathcal{X}\times\mathcal{Y}$ is…
user52534
  • 751
3
votes
1 answer

prove the space is Polish

Suppose $X=A\cup B$ is a separable metrizable space and $X$ is Baire. If $A$ is an open Polish subspace of $X$, $B$ is countable and closed nowhere dense in $X$, $A\cap B=\emptyset$, then is $X$ a Polish space?
aha
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