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Results tagged with descriptive-set-theory
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user 15002
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.
5
votes
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When can I "draw" a topology in Baire space?
I have two things to offer, the first of which could help with getting better characterizations, the latter should give ample of examples.
Since I am not aware of standard terminology, call $(X,\tau' …
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Is there a standard Borel space of finitely branching real trees?
A natural way to represent a finitely branching tree over $\mathbb{R}$ is to separate the structure of the tree from the content (ie its labels from $\mathbb{R}$).
We can describe the structure of the …
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5
votes
Picking a real for every non-empty open set in $\mathbb{R}$
We can get a rather simple such function $f$. If if we measure this in terms of descriptive set theory, we get a Baire class 1 function (we need DST for Quasi-Polish spaces here, as $\mathcal{E} = \ma …
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5
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Which topological spaces have a standard Borel $\sigma$-algebra?
Here are two examples showing that none of your candidate notions work.
First, we can observe that every Quasi-Polish space (https://doi.org/10.1016/j.apal.2012.11.001) admits a Baire class 1 isomorph …
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1
vote
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Is the set of clopen subsets Borel in the Effros Borel space?
Here is a negative answer for $\mathbb{N}^\mathbb{N}$.
Given a countably-branching tree $T$, we built a new countably-branching tree $T'$ in two steps. First, for any $\sigma \in T$ we place $(\sigma …
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2
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Hausdorff quasi-Polish spaces
Some further examples of Hausdorff Quasi-Polish but not Polish spaces can be found in
Kihara, Ng & Pauly: Enumeration degrees and non-metrizable topology arXiv 1904.04107
In Example 3.17, we construct …
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6
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Is every path connected space continuously path connected
As pointed out by Anton Petrunin, the condition is stated is equivalent to the space being contractible. Following Omar Antolín-Camarena, this can be seen since $(t,x) \mapsto f(t,x,y_0)$ constitutes …
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3
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Can we inductively define Wadge-well-foundedness?
Long comment:
It is difficult to answer "no" to the main question, given the unlimited potential interpretations of "reasonable". Still, I would be very surprised by a positive answer.
The reason is …
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5
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Can these alternating series games be undetermined?
If we take $s_i = 2^{-i}$, we should even get that $\mathrm{ZFC}$ proves the existence of some $\mathcal{X}$ with undetermined $A_{(2^{-i})_{i \in \mathbb{N}}}(\mathcal{X})$. The key parts are that di …
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Examples of Baire Class $\xi+1$ but not $\xi$ functions for each countable ordinal $\xi.$
A somewhat concrete example of function which is Baire class $\zeta$ but not Baire class $\gamma$ for any $\gamma < \zeta$ is the $\zeta$-th Turing jump. This is essentially the iterated version of Sh …
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6
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An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
Yes, AC gives us a continuums-sized measure 0 set without the perfect set property.
For the construction of just any continuums-sized subset of $\mathbb{R}$, what matters is that $\mathbb{R}$ has card …
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9
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Is every compact, sober, second-countable space the image of $2^\omega$?
We can built a counterexample by adding a bottom element to $\mathbb{N}^\mathbb{N}$. Let $\mathbb{N}^\mathbb{N}_\bot$ have the underlying set $\mathbb{N}^\mathbb{N} \cup \{\bot\}$, and let a set be op …
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What is known about these "explicitly represented" spaces?
Your notion seems to be what would be called a $2^\omega$-based represented space following de Brecht, Schröder and Selivanov here (but there might be a misaligned in the details somewhere):
https://c …
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