Articles of descriptive set theory

Does a dense $G_\delta$ subset of a complete metric space without isolated points contain a perfect set?

Let $(X,d)$ be a complete metric space without isolated points. Is it true that each dense $G_\delta$ subset of $X$ contains a nonempty perfect set (i.e. closed without isolated points)? Thanks.

Cantor-Bendixson rank of a closed countable subset of the reals, and scattered sets

I am reading the notes in the following link, and I am a bit unclear about the connection between scattered sets, countable sets, and sets for which the Cantor-Bendixson derivative is eventually the empty set. On page 3, the author says: A closed set X (of the reals) is scattered if $X^{\alpha}=\emptyset$, where $\alpha$ […]

Is there a name for a topological space $X$ in which very closed set is contained in a countable union of compact sets?

Is there a name for a topological space $X$ which satisfies the following condition: Every closed set in $X$ is contained in a countable union of compact sets Does Baire space satisfy this condition? Thank you!

Quotient space and Retractions

I’m trying to learn something about topology and category theory. Let us consider the category of compact Polish spaces. The category contains all quotients of all objects (wikipedia) For an equivalence relation $\sim$ on $X\times X$, we denote with $X/\!\sim$ the quotient space and with $q:X\rightarrow X/\!\sim$ the quotient map. I’m trying to convince myself […]

Dense subset of the plane that intersects every rational line at precisely one point?

It seems there should exist a non-measurable bijection $f: \mathbb{R}\rightarrow \mathbb{R}$. And thus we can obtain a non-measurable graph on $\mathbb{R}^2$ which intersects every horizontal or vertical line at precisely one point. Does anyone know how to “construct” such a map? Can it be further made into a automorphsim (w.r.t the addtive group or field […]

Conditional probability independent of one variable

Let $X,Y$ and $Z$ be Borel spaces (that is, Borel subsets of Polish spaces) and let $\mathcal P(X)$ denote the Borel space of all Borel probability measures on $X$. For a product measure $P\in \mathcal P(X\times Y\times Z)$ let $\kappa:X\times Y\to\mathcal P(Z)$ be a regular conditional probability on $Z$ given $X\times Y$. We say that […]

Why the set of outcomes generated by a fixed strategy of one player in Gale-Stewart game is a perfect set?

In the proof that there is a payoff set $X$ such that the Gale-Stewart game is not determined(see here, Proposition 3.1.). I don’t know why $X$, the set of all outcomes generated by a fixed strategy of one player, constitutes a perfect set. I can see it’s true, if the fixed strategy is a constant […]

Is every $F_{\sigma\delta}$-set a set of points of convergence of a sequence of continuous functions?

It is well known that if $\langle f_n:n\in\mathbb{N}\rangle$ is a sequence of continuous functions, $f_n\colon\mathbb{R}\to\mathbb{R}$, then $\big\{x\in\mathbb{R}:\lim_{n\to\infty}f_n(x)\text{ exists}\big\}$ is an $F_{\sigma\delta}$-set (see this post). I am asking if the converse is true, i.e., whether for every $F_{\sigma\delta}$-set $E\subseteq\mathbb{R}$ there exists a sequence $\langle f_n:n\in\mathbb{N}\rangle$ of continuous functions, $f_n\colon\mathbb{R}\to\mathbb{R}$, such that $\big\{x\in\mathbb{R}:\lim_{n\to\infty}f_n(x)\text{ exists}\big\}=E$. My attempt: I […]

Measure zero on all Fat Cantor Sets

Let $F_n\subset [0,1]$ be a Fat Cantor Set (so that $[0,1]\setminus F_n$ is dense) of Lebesgue measure $1 – 1/n$, and let $F = \bigcup_n F_n$. Does there exist a probability measure $\mu$ on $[0,1]$ such that $\mu(F + x) = 0$ for all $x\in [0,1]$? Here $+$ is a cyclic shift, so $0.5 + […]

The collection of all compact perfect subsets is $G_\delta$ in the hyperspace of all compact subsets

Let $X$ be metrizable (not necessarily Polish), and consider the hyperspace of all compact subsets of $X$, $K(X)$, endowed with the Vietoris topology (subbasic opens: $\{K\in K(X):K\subset U\}$ and $\{K\in K(X):K\cap U\neq\emptyset\}$ for $U\subset X$ open), or equivalently, the Hausdorff metric. We want to show that $K_p(X)=\{K\in K(X):K \text{ is perfect}\}$ is $G_\delta$ in $K(X)$. […]