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Hot answers tagged descriptive-set-theory

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A rare property of Hausdorff spaces

Yes, there is such a space. Let $X=2^{\omega_1}$ be the space of binary sequences of length $\omega_1$, in the order topology generated by the lexical order. So $X$ consists of the branches through ...
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Is every real number in [0,1] a product of three (or more) Cantor set's numbers?

Yes, every real number $u \in [0,1]$ can be written as $u = x^2 y$ where $x,y \in C$ are in the Cantor set $C$. In particular, every real number in $[0,1]$ is a product of three Cantor set elements. ...
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A rare property of Hausdorff spaces

How about this example: $[0,1]^A$ with the product topology, where $A$ is uncountable. Then every nonempty $G_\delta$ set is uncountable. Since your sets $f^{-1}(x)$ are $G_\delta$ sets, this has ...
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Closed balls vs closure of open balls

The following theorem (or its corollary) implies negative answer to the original question. Theorem. For any point $x$ of a metric space $(X,d)$ the set $R_x:=\{r>0:cl(B(x,r))\ne \bar B(x,r)\}$ ...
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Writing a function on $\mathbb{R}$ as a sum of two injections

The answer is yes. Every function on the reals is the sum of two injective functions, and this can be done in a highly effective manner, constructing the two functions $g,h$ from $f$ without any need ...

Quantifier complexity of the definition of continuity of functions

It is truly a very nice question, one of those questions with an answer one feels must be right, but it is not so clear at first how to prove it. Nevertheless, aiming at partial progress, I claim that ...
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How (non-)computable is set theory?

The question is extremely interesting, and I have looked into this kind of thing with various colleagues (including Russell Miller and Kameryn Williams), although our investigation has not yet ...
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Is there a Borel subset of $\mathbb{R}^{2}$, with finite vertical cross-sections, whose projection onto the first component is non-Borel?

No, no such set exists. This is a special case of the Lusin–Novikov theorem; see e.g. Kechris, Classical Descriptive Set Theory, Theorem 18.10. In general, let $X,Y$ be standard Borel spaces, ...
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Is the set of all ICC amenable groups countable?

The set of countable ICC groups doesn't exist, so I think you're asking about isomorphism classes. There are continuum many non-isomorphic locally finite fields (e.g., take, for $S$ any set of primes, ...
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How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?

The answer to the third question is yes. This is due to Kechris, who showed that $\text{ACA}_0$ plus schematic PD implies schematic DC. This appears in the last section of "The Axiom of ...
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Uncountable disjoint closed coverings of $[0,1]$

This is question has a long and interesting history, which is discussed in Arnie Miller's paper cited below. The first construction of a model of ZFC + $\aleph_1 < 2^{\aleph_0}$ where $[0,1]$ can ...
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Is the set of all ICC amenable groups countable?

As for the second question, the equivalence relation on the class of countable ICC groups given by $G \sim \Gamma$ if and only if $L(G) \cong L(\Gamma)$ is very interesting and usually called $W^*$-...
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Partitions of the real line into Borel subsets

The answer to both problems is no! If the Cohen forcing is used to add lots of reals to a countable transitive model of GCH, then in the resulting extension, any partition of the real line into Borel ...
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Detecting uncountable cardinals in $(\mathbb{R};+,\times,\mathbb{N})$

For the first question (distinct regular cardinals $>\aleph_1$): Force ZFC + MA + $2^{\aleph_0}=\aleph_3$ over $L$ in the usual way (see Jech, Theorem 16.13; note the forcing is ccc and it forces ...
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Conflating reals and sets of countable ordinals "nicely"

The technique of almost disjoint forcing was introduced in MR0289291 (44 #6482). Jensen, R. B.; Solovay, R. M. Some applications of almost disjoint sets. In Mathematical Logic and Foundations ...
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• 72.4k

Writing a function on $\mathbb{R}$ as a sum of two injections

It works at least for (locally) absolutely continuous functions. Such a function is the integral of a locally $L^1$ function. This weak derivative can be written as a sum of a positive and negative ...
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How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?

Here is a natural model $M$ of $Z_2$ where projective DC fails. Starting with a model of ZFC + $V=L$, force over $L$ with the Levy collapse $\mathrm{Coll}(\omega,{<\aleph_\omega})$ collapsing all ...
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Ordinal-definable witnesses to the perfect set property?

Let $A$ be the set of those reals that are not OD. Then $A$ is OD (since I've just defined it), and it's uncountable (since its complement, being a well-orderable set of reals, must be countable ...
• 72.4k
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What is the descriptive complexity of a set added by Cohen forcing?

Since Cohen forcing is weakly homogeneous, all hereditarily ordinal definable sets in the Cohen extension are already in the ground model. That applies in particular to any ordinal definable real, ...
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Is $V=L$ equivalent to there being a $\Sigma_1$ well-ordering of the universe?
No, see the paper On $Σ_1$ Well-Orderings of the Universe by Harrington and Jech.