Tagged Questions

In descriptive set theory we mostly study Polish spaces such as the Baire space, the Cantor space, and the reals. Questions about the Borel hierarchy, the projective hierarchy, Polish spaces, infinite games and determinacy related topics, all fit into this category very well.

49 views

A property of some Hausdorff topological spaces

Let $X$ be a Hausdorff topological space such that any closed subset of $X$ with empty interior is finite. show that $X$ has an isolated point.
125 views

102 views

Filter, which does not have the Baire property

I do not understand the following proof of the following theorem. It is part of Theorem 4.1.2 (Talagrand theorem) in the book Tomek Bartoszyński and Haim Judah: Set theory. On the structure of the ...
42 views

Definition of length of a sequence

In definition $3.11$, the author defined the following rank: If the sets $A$ and $B$ can be separated by a transfinite difference of closed sets, then let $\alpha_1(A,B)$ denote the length of ...
104 views

Meager subset of $2^\omega$

Suppose we do have a filter $\mathcal{F}$ on $\omega$ which contains the cofinite filter, so $X\in\mathcal{F}$ implies $X$ is infinite. For $X\in\mathcal{F}$, let $f_X$ be the increasing enumeration ...
29 views

Does this system of open sets have to cover the whole space?

I have been studying basics of descriptive set theory lately. In the lecture notes I follow (sadly, the notes are written in Czech), there is the following definition: Let X be a topological space....
21 views

How to prove that these sets are $\mathbf{\Delta}_\alpha^0$.

I have been studying basics of descriptive set theory lately. In the lecture notes I follow (sadly, the notes are written in Czech), there is a lemma (which is used to prove that the functions of ...
20 views

Why for every rational $p$ the level sets $\{ f \leq p \}$ and $\{ f \geq p \}$ are $\prod_{\xi+1}^0$ sets?

$f$ is a real-valued baire class $\xi$ function if $\{ f < c \}$ and $\{ f >c \}$ are in $\sum_{\xi +1}^0$ for every $c \in \mathbb{R}$. In the proof of Proposition $5.2$ page $24$, we have ...
26 views

113 views