Topological tameness beyond the Gandy-Harrington topology

Question

The Gandy-Harrington topology on $\omega^\omega$ is the topology generated by all lightface $\Sigma^1_1$ sets; that is, all sets which are continuous-in-the-usual-sense images of $\omega^\omega$.

Although this topology is definitely less nice than the standard one - for example, it is non-metrizable - it satisfies the strong Choquet property: this is the statement that player I has a winning strategy in a certain topological game. From the strong Choquet property we can deduce many results whose proof would follow easily from metrizability, so in some sense the Gandy-Harrington topology is "close enough" to metrizable. One particularly nice result we can show is: if $A$ is a nonempty lightface $\Sigma^1_1$ subset of $\omega^\omega$, then $A$ has an element $x$ such that $\mathcal{O}^x\equiv_T\mathcal{O}\oplus x$.

A natural question now is to ask about topologies generated by lightface sets further up the projective hierarchy; e.g., let $\tau_k$ be the topology on $\omega^\omega$ generated by lightface $\Sigma^1_k$ sets. And similarly, we can define $\tau_\Gamma$ for any pointclass $\Gamma$ (although we're probably only interested in, say, Spector pointclasses). Unfortunately, for $k>1$ the strong Choquet property fails badly in $\tau_k$!

My question is:

Is there a weakening of the strong Choquet property that holds for $\tau_2$? (Or more generally for $\tau_\Gamma$ for nice enough pointclass $\Gamma$.)

Relatedly, for a real $x$ let $\mathcal{O}_2^x$ be the set of $x$-computable codes for nonempty $\Sigma^1_2$ sets, and let $\mathcal{O}_2=\mathcal{O}_2^\emptyset$.

Is it the case that every lightface $\Sigma^1_2$ subset of $\omega^\omega$ contains an $x$ with $\mathcal{O}_2^x\equiv_T\mathcal{O}_2\oplus x$?

I've tagged "set-theory" and "inner-model-theory" because it seems reasonable to me that answers might depend on structural hypotheses about the universe; feel free to remove either tag if this seems bonkers.

Answer 1

Related to the second question: Every non-empty $\Sigma^1_2$ subset of $\omega^\omega$ has a $\Delta^1_2$ element $x$, namely its $L$-least member. The complete $\Sigma^1_2$ subset of $\omega$ relative to this $x$ is $\Sigma^1_2$, so $\mathcal{O}_2^x$ is recursively isomorpic to $\mathcal{O}_2$.

Answer 2

As mentioned by Ted, the appropriate generalization is for odd levels only. In Hjorth "Variations of the Martin-Solovay tree" and "Some applications of coarse inner model theory", the following folklore result that generalizes Harrington's proof of Silver's dichotomy is stated:

Assume boldface $\bf\Delta^1_2$ determinacy. If $E$ is a thin $\Pi^1_3$ equivalence relation on $\mathbb{R}$, then for every $x$, there is a $\Delta^1_3(<u_\omega)$ set $A$ such that $x \in A \subseteq [x]_E$.

As defined in these papers, for a fixed $\alpha<u_\omega$, $A\subseteq \mathbb{R}$ is $\Sigma^1_3(\alpha)$ means there is $B\subseteq \mathbb{R}^2$ such that $x\in A$ iff $(x,y)\in B$ for some sharp code $y$ for $\alpha$. $\Sigma^1_3(<u_\omega)$ means $\Sigma^1_3(\alpha)$ for some $\alpha<u_\omega$. $\Delta$-pointclasses are defined in the obvious way.

The correct generalization of the Gandy-Harrington topology appears to be generated by $\Sigma^1_3(<u_\omega)$ sets. Also as a corollary, every thin $\Pi^1_3$ equivalence relation is $\Delta^1_3$ reducible to a $\Pi^1_3$ equivalence relation on a $\Pi^1_3$ subset of $u_\omega$ (in the sharp codes); every thin $\Delta^1_3$ equivalence is $\Delta^1_3$ reducible to equality on $u_\omega$.

I haven't read the proof of this generalization. So it would be nice if someone can expand a bit more. I also don't know the proof of the basis theorem at lower level with this approach, even though it seems to be straightforward.