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Results tagged with descriptive-set-theory
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user 1946
Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
10
votes
Accepted
Relation between projective hierarchy and universally measurable sets
Under the continuum hypothesis, or even just when
$2^{\aleph_0}<2^{\aleph_1}$, there must be universally measurable
sets that are not projective. The reason is that Hausdorff proved
that there are alw …
- 236k
7
votes
Accepted
Equality of Borel sets
Nice question!
Unfortunately, this relation is not Borel. Indeed, it is $\Pi^1_1$-complete,
even at very low levels of the hierarchy.
To see this, suppose that $x$ is a real coding a binary relatio …
- 236k
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 th …
- 236k
9
votes
Accepted
Do the Lebesgue-null sets cover "all the sets can naturally be regarded as sort-of-null sets"?
The answer is no, by a construction using the axiom of choice.
We shall build a counterexample set $A$ by a transfinite recursive
process of length continuum. At each stage, we shall promise that
cer …
- 236k
8
votes
Games that never begin
This is a very nice question.
Observation 1. Some strategies have no play that accords
with them. Consequently, such a strategy for Alice is winning in
any game, since every play in conformance with …
- 236k
6
votes
Connectedness of the complement of small subsets (extended question)
Every countable subset $A\subset \mathbb{R}^n$, with $n\gt 1$, and indeed, every subset of size less than the continuum, has a path-connected complement. This is because for any two points in the comp …
- 236k
15
votes
Accepted
Wanted: chain of nowhere dense subsets of the real line whose union is nonmeagre, or even co...
Theorem. There is no chain of nowhere dense subsets of $\mathbb{R}$ whose union contains an interval.
Proof. Suppose there was such a chain $\{\ B_i \mid i\in I\ \}$, where $\langle I,\lt\rangle$ is …
- 236k
28
votes
Accepted
The reals as continuous image of the irrationals
For any irrational number $x$, let $f(x)$ be the real number arising from the integer part of $x$, together with every other digit of the rest of the expansion of $x$.
This is surjective, since one …
- 236k
6
votes
Accepted
Is the set of subsequences of branches through a tree Borel?
With your assumption that the tree consists of increasing
sequences only, then the answer is yes, this is Borel. The reason
is that we can identify whether or not $x$ is a subsequence of a
branch thro …
- 236k
10
votes
Accepted
Is it inconsistent for a model of set theory to contain its own first order theory?
First, let me point out as the others have that if there are large
cardinals, then indeed we expect this situation. For example, if
there is a worldly cardinal, a cardinal $\kappa$ for which $V_\kappa …
- 236k
5
votes
Cofinality of a $\sigma$-ideal of $\mathbb{R}$
If the continuum hypothesis holds, then there is such an ideal. Indeed, we need only to assume that $2^\omega\lt
2^{\omega_1}$, a weakening of CH.
Consider the tree $T=2^{\lt\omega_1}$ of all binary …
- 236k
4
votes
Accepted
Analytic enlargement of an analytic set
The answer is no, not necessarily. It can happen that there is no such set $B$.
A counterexample is provided whenever $A$ is a Borel subset of the plane, while the projection $\pi(A)$ is analytic, b …
- 236k
11
votes
Accepted
cardinality of perfect sets in generalized Baire space
It is consistent to have a perfect set of size
$\kappa$, or of intermediate size between $\kappa$ and $2^\kappa$.
To see this, suppose that $\kappa$ is an inaccessible cardinal in $V$, and let
$T=2^ …
- 236k
4
votes
Given a sequence of reals, we can find a dense sequence avoiding it, but can we find one con...
If you replace the reals $\mathbb{R}$ with Cantor space $2^{\mathbb{N}}$ or with Baire space $\mathbb{N}^{\mathbb{N}}$ (homeomorphic to the space of irrationals), then the answer is yes. Indeed, one c …
- 236k
6
votes
Accepted
Does the boldface class $\Delta^1_2$ have the uniformization property? (assuming $V=L$)
Unless I am mistaken, it seems to me that $\Delta^1_2$ does have the uniformization property in $L$.
For any set $A$ in $\Delta^1_2$, let $B$ select the $L$-least witness on each slice. So $B$ unifo …
- 236k