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Results tagged with descriptive-set-theory
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user 22277
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.
6
votes
Accepted
Is it possible for a separable metric and a non-separable metric to have the same Borel $\si...
It is possible that $(X,d_{1})$ has the same Borel sets as $(X,d_{2})$ when $d_{1}$ is separable and $d_{2}$ is not by assuming the Martin's axiom and the negation of the continuum hypothesis by the a …
3
votes
A rare property of Hausdorff spaces
If $(X,\mathcal{S})$ is a regular space, then the $P$-space coreflection of $(X,\mathcal{S})$ is the space generated by the basis $\{\bigcap_{n\in\omega}U_{n}\mid\forall n\,U_{n}\in\mathcal{S}\}$. A c …
8
votes
Accepted
Cofinality of a $\sigma$-ideal of $\mathbb{R}$
$\textbf{A counterexample when $2^{\aleph_{0}}$ is regular}$.
This holds if $2^{\aleph_{0}}$ is a regular cardinal. In fact, it holds for any regular cardinal. If $\kappa$ is a regular cardinal, then …
4
votes
Accepted
Does every separated measurable space embed into a power of $\{0,1\}$?
Let $X$ be a set and give $X$ the discrete topology. Give $X\cup\{\infty\}$ the one-point compactification $X$. Then every subset $U$ of $X$ is open in $X\cup\{\infty\}$. Therefore, every subset of $U …
3
votes
Accepted
Countably generated $\sigma$-algebra
For the first question, the answer is yes. There is an isomorphism between measure algebras. Let $B$ be the Boolean algebra of all measurable sets modulo the collection of null sets. Then define a met …
4
votes
Accepted
Different Metrics for Baire Space and their induced Topologies
I am going to answer the question that you all ask whenever you see a metric space "is it complete?"
$\mathbf{Proposition}$ The metric $d'$ on $\mathbb{N}^{\mathbb{N}}$ is not a complete metric.
$\ …
14
votes
Accepted
Generalizations of the Tietze extension theorem (and Lusin's theorem)
There is a nice characterization of the spaces $X$ where the Tietze extension theorem holds for all complete separable metric spaces $Y$. We say that a Hausdorff space $X$ is ultranormal if whenever $ …
12
votes
Accepted
Is there a suitably generalized Baire property for topological spaces of arbitrary cardinali...
I am not sure if this is the kind of answer you were looking for, but since no one has given an answer yet, I think it is a good idea to say what little I know about larger cardinal analogues of the B …
1
vote
Cohen algebra (generalization)
I was skimming through the handbook of set theory, and I stumbled upon a combinatorial characterization of the Cohen algebras $Ro(2^{\kappa})$ where $\kappa$ is any infinite cardinal and $2^{\kappa}$ …
3
votes
Accepted
Definition of $F_{\sigma}$ sets in terms of $\varepsilon$?
Suppose that $(X,d)$ is a complete metric space. Then a subset $G\subseteq X$ is a $G_{\delta}$-set precisely when $G$ can be given a complete metric which induces the subspace topology on $G$. The no …
5
votes
Accepted
Topological analog of the Lusin-N property
Let $K$ be a nowhere dense closed subset of $[0,1]$ of positive Lebesgue measure $\delta>0$. Such a set $K$ can be obtained using the standard technique for constructing a nowhere dense closed set of …
4
votes
Cohen algebra (generalization)
There is a characterization of what you call a random algebra found in [1][Ch. 15. Sec 3]. This characterization involves the notion of a measure algebra.
We define a measure algebra to be a Boolean …