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Results tagged with descriptive-set-theory
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user 112417
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.
4
votes
1
answer
140
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Hedgehog of spininess $κ$ is an absolute retract?
Let $κ$ be an infinite cardinal, $S$ a set of cardinality $κ$, and let
$I = [0, 1]$ be the closed unit interval. Define an equivalence
relation $E$ on $I × S$ by $(x,α) E (y,β)$ if either $x = 0 = y$
…
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3
votes
0
answers
77
views
What is the name of the (possibly well-known) class of $\pi$-monolithic compact spaces?
A compact space $X$ is called ${\it \pi-monolithic}$ if whenever a surjective continuous mapping $f:X\rightarrow K$ where $K$ is a compact metric space there exists a compact metric space $T\subseteq …
- 1,101
4
votes
1
answer
103
views
Are there such a complete metric space X of weight k (w(X)=k) and ....?
Are there such a complete metric space $X$ of weight $k<\mathfrak{c}$ ($w(X)=k$) and a family $\{F_{\alpha}: \alpha<k\}$ of closed subsets of $X$ that $k<|X\setminus \bigcup F_{\alpha}|<\mathfrak{c} …
- 1,101
7
votes
1
answer
373
views
What is an example of a meager space X such that X is concentrated on countable dense set?
A topological space $X$ is concentrated on a set $D$ iff for any open set $G$ if $D\subseteq G$, then $X\setminus G$ is countable.
What is an example of a separable metrizable (uncountable) meager (me …
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4
votes
0
answers
136
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Is there a condensation of a closed subset of $\kappa^\omega$ onto $\kappa^\omega\setminus A...
Let $\aleph_1\le\kappa<c$ and $A\subset \kappa^{\omega}$ such that $\lvert A\rvert\le\kappa$.
Is there a condensation (i.e. a bijective continuous mapping) of a closed subset of $\kappa^\omega$ onto …
- 1,101
3
votes
1
answer
125
views
What are the names of the following classes of topological spaces?
The closure of any countable is compact.
The closure of any countable is sequentially compact.
The closure of any countable is pseudocompact.
The closure of any countable is a metric compact set.
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7
votes
1
answer
207
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Are σ-sets preserved by Borel isomorphisms?
Recall that a $\sigma$-space is a topological space such that every $F_{\sigma}$-set is $G_{\delta}$-set.
$X$ - $\sigma$-set, if $X$ is a $\sigma$-space and it is subset of real line $R$.
Let $F$ …
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2
votes
0
answers
163
views
Is there a Lusin space $X$ such that ...?
Is there a Lusin space (in the sense Kunen) $X$ such that
$X$ is Tychonoff;
$X$ is a $\gamma$-space ?
Note that if $X$ is metrizable and a $\gamma$-space then it is not Lusin.
In mathematics, a Luz …
- 1,101
3
votes
0
answers
141
views
Which cardinal $\kappa\geq \omega_1$ is critical for the following property...?
Which cardinal $\kappa\geq \omega_1$ is critical for the following property:
Let $X\subset \mathbb R$ and $\kappa>|X|\geq \omega_1$. Then there is an uncountable family $\{X_{\alpha}\}$ such that $|X …
- 1,101
3
votes
1
answer
174
views
Is there a metric separable space with the following properties...?
Let $\omega_1<\mathfrak{q}_0$ where $\mathfrak{q}_0:=\min\{|Y|:Y\subseteq \mathbb{R}$, $Y$ is not a $Q$-space$\}$.
Is there a metric separable space $X$ with the following properties:
$|X|\geq\mathfr …
- 1,101
6
votes
1
answer
145
views
Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ ...
Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?
- 1,101
4
votes
1
answer
94
views
Is there a hereditary $\sigma$-space $X$ such that it is not $Q$-space?
A topological space $X$ is called a $\sigma$-space if every $F_{\sigma}$-subset of $X$ is $G_{\delta}$.
A topological space $X$ is called a $Q$-space if any subset of $X$ is $F_{\sigma}$.
Definition. …
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10
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0
answers
495
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Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?
Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?
Here $\mathfrak p$, $\mathfrak b$, $\mathfrak q$ are small uncountable cardinals:
$\mathfrak p$ is the smallest ca …
- 1,101
4
votes
0
answers
126
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An uncountable Baire γ-space without an isolated point exists?
An open cover $U$ of a space $X$ is:
• an $\omega$-cover if $X$ does not belong to $U$ and every finite subset of $X$ is contained in a member of $U$.
• a $\gamma$-cover if it is infinite and each $x\ …
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5
votes
1
answer
369
views
Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$?
Recall that
$\mathfrak{p}=\min\{|F|: F$ is a subfamily of $[\omega]^{\omega}$ with the sfip which has no infinite pseudo-intersection $\}$.
The cardinal $\mathfrak{q}_0$ defined as the smallest cardi …
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