# Tagged Questions

36 views

### Sets that are not $\infty$-Borel

I have seen a few techinques for proving that certain sets of real numbers are $\infty$-Borel (definition) but it just occurred to me that I don't know of any way to prove that a set of real numbers ...
110 views

### Bijection between closed uncountable sets and R?

This is probably a really stupid question but I'm hoping it's true: can we find a bijection between every closed uncountable set and R (real number line)?
60 views

132 views

### $\bf AD$ implies $\bf AC_{\omega}(\mathbb{R})$?

How to show that $\bf AD$ implies $\bf AC_{\omega}(\mathbb{R})$? $\bf AD$ is abbreviated for axiom of determinacy. $\bf AC_{\omega}(\mathbb{R})$ states that for each family $(X_i)_{i∈\omega}$, in ...
105 views

423 views

### “Nice” well-orderings of the reals

I have a question which I believe could be easily resolved if I happened to look at the right source - hence my asking it here as opposed to at MathOverflow. I've tried googling it, but I haven't been ...
149 views

### Does there exist a subset of $\mathbb{R}$ of cardinality $2^{\aleph_0}$ that has no perfect subset?

Assuming the axiom of choice, is there a way to construct a subset of the reals of cardinality $2^{\aleph_0}$ that has no perfect subset?
3k views

### Cardinality of Borel sigma algebra

It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
247 views

### Separable metric space with 0-dimensional kernel

Suppose $X$ is a separable metric space, let $D(X)$ denote the Cantor-Bendixson derivative of $X$, and $D_\alpha(X)$ the $\alpha$-th derivative of $X$. We denote $\operatorname{Ker}(X)$ the kernel of ...
192 views

### $\Sigma_{\alpha+1}^0(X)$ universal set based on the Baire space

In my attempts to write a cleaner proof (compared to what I was taught) to the fact that for uncountable Polish spaces the Borel hierarchy does not stop until $\omega_1$ I am trying to prove the ...
185 views

### Sets invariant under sections

Let $X$ be a compact in the Polish space (metric, complete, separable) and $G\subseteq X\times X$ is open. For $x\in X$ we define the section of $G$:  s(x) = \{y\in X|\langle x,y \rangle \in ...
263 views

### Universal sets in metric spaces

Today I saw in the class the theorem: Suppose $X$ is a separable metric space, and $Y$ is a polish space (metric, separable and complete) then there exists a $G\subseteq X\times Y$ which is open and ...
In most Descriptive Set Theory books, the rationale for working with the Baire space ($\mathbb{N}^{\mathbb{N}}$) as opposed to the real line ($\mathbb{R}$) is that the connectedness of the latter ...