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35 votes
Accepted

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 ...
Joel David Hamkins's user avatar
32 votes
Accepted

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. ...
Zach Teitler's user avatar
  • 6,227
29 votes

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 ...
Gerald Edgar's user avatar
  • 40.6k
28 votes
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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 $R_x:=\{r>0:cl(B(x,r))\ne \bar B(x,r)\}$ ...
Taras Banakh's user avatar
  • 41.1k
27 votes
Accepted

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 ...
Joel David Hamkins's user avatar
26 votes

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 ...
Joel David Hamkins's user avatar
25 votes
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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 ...
Joel David Hamkins's user avatar
20 votes
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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, ...
Nate Eldredge's user avatar
20 votes
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Can an injective $f: \Bbb{R}^m \to \Bbb{R}^n$ have a closed graph for $m>n$?

There is no such function. Suppose $f: \mathbb R^m \rightarrow \mathbb R^n$ is an injective function with $\Gamma_f$ closed in $\mathbb R^{m+n}$. For each $i \in \mathbb N$, let $K_i = f^{-1}([-i,i]^...
Will Brian's user avatar
  • 17.8k
20 votes
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Homeomorphisms and "mod finite"

Define $f : C \to C$ by the formula $$ f(x) = x_0 \cdot (x \oplus \sigma(x)) $$ where $\cdot$ is word concatenation, $\oplus : C \times C \to C$ is coordinatewise xor, and $\sigma(x)_i = x_{i+1}$ is ...
Ville Salo's user avatar
  • 6,457
19 votes
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Topological proof that a Vitali set is not Borel

Sometimes a convenient substitute for Lebesgue measurability is the property of Baire. Just like Lebesgue measurability, the class of sets with this property is a $\sigma$-algebra containing the open ...
Wojowu's user avatar
  • 27.7k
19 votes

Examples of concrete games to apply Borel determinacy to

The game of infinite Hex, proceeding from an arbitrary position, is a good example with all the features you seek. The game was the subject of my Oxford student Davide Leonessi's masters MFoCS ...
Joel David Hamkins's user avatar
18 votes
Accepted

Is every element of $\omega_1$ the rank of some Borel set?

In $\mathsf{ZFC}$ the Borel hierarchy of an uncountable Polish space $X$ has length exactly $\omega_1$, meaning that $\mathbf{\Sigma}^0_\xi(X)\neq\mathbf{\Pi}^0_\xi(X)$ for all $\xi<\omega_1$. The ...
Alessandro Codenotti's user avatar
16 votes
Accepted

Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ with injective restriction $f|\mathbb Q^\omega$?

It looks like no. Assume the contrary. We may start with two distinct rationals $q_1,p_1$ such that the sets $f(q_1\times \mathbb{R}^{\omega-1})=f(\{(q_1,\cdot,\cdot,\dots)\})$ and $f(p_1\times \...
Fedor Petrov's user avatar
15 votes

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, ...
YCor's user avatar
  • 62.3k
15 votes
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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 ...
Gabe Goldberg's user avatar
14 votes
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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 ...
François G. Dorais's user avatar
14 votes

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 $W^*$-...
Stefaan Vaes's user avatar
  • 4,226
14 votes
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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 ...
Will Brian's user avatar
  • 17.8k
14 votes
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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 ...
Farmer S's user avatar
  • 9,167
13 votes
Accepted

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 ...
Andrés E. Caicedo's user avatar
13 votes
Accepted

Can the Turing degrees be linearly ordered?

You can't linearly order the Vitali ($\mathcal{P}(\omega)/\mathrm{Fin}$) degrees if every set of reals has the property of Baire, since you can't even choose between complementary degrees. The set of $...
Paul Larson's user avatar
  • 2,520
13 votes
Accepted

The Axiom of Determinacy and the Banach-Mazur game

The claim is false. The Banach-Mazur game, also known as the $**$-game shows (and is equivalent to) that every set of reals has the Baire property. What is true, as you've noted, is that if one has a ...
Rachid Atmai's user avatar
  • 3,754
13 votes
Accepted

Do Borel subsets of the plane with null sections have Borel projections?

Notice that $\omega^\omega$ can be embedded to a null subset of itself by sending any sequence $(a_0,a_1,a_2,\dots)$ to $(a_0,0,a_1,0,a_2,0,\dots)$. So any Borel phenomenon that can happen in $\omega^\...
Andreas Blass's user avatar
13 votes

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 ...
Joonas Ilmavirta's user avatar
13 votes

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 ...
Farmer S's user avatar
  • 9,167
12 votes
Accepted

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 ...
Andreas Blass's user avatar
12 votes
Accepted

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, ...
Andreas Blass's user avatar
12 votes
Accepted

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.
Mohammad Golshani's user avatar

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