Descriptive set theory

Where does $\Q$ sit in the Borel hierarchy of $\R$?

Which of these spaces (with their usual topologies) are Polish? Choose all that apply.

Which kind of subset of $\R^2$ can have a projection onto the $x$-axis that is not Borel?

In the proof that an uncountable analytic set $A = f[\mathcal N]$ contains a perfect set, why do we first remove the open set $U$ (the union of the $N_s$ with $f[N_s]$ countable)?

Let $X = \{0\} \cup \{\tfrac1n : n \ge 1\} \cup \{\tfrac1m + \tfrac1n : m, n \ge 1\}$, a closed subset of $\R$. The derived set $X'$ removes the isolated points. How many times must the derivative be applied before the empty set is reached (the Cantor–Bendixson rank of $X$)?

Mazurkiewicz (1936) studied $D = \{f \in C[0, 1] : f \text{ is differentiable at every point}\}$, where $C[0, 1]$ has the sup norm. Which statement is correct?

Which Lean statement formalises (one direction of) Suslin's theorem in a Hausdorff space with Borel σ-algebra?

The Lean proof of rat_not_isGδ applies Dense.inter_of_Gδ to two sets. What does it conclude, and why is that a contradiction?

A tree on $\{0, 1\}$ is given by a predicate tree(s) on binary strings. Write split_height(tree, n): the height of the largest full binary tree whose branching nodes are branching nodes of the tree, within depth $n$. Recursively, at a node $t$ of length $n$ the height is $0$; otherwise it is the maximum of the children's heights and, if both children are in the tree, $1 + \min$ of their heights.