Define a pointclass to be:


* *boldface inductive-like* if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions, and
* *lightface inductive-like* if it is $\omega$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions.


Given a lightface inductive-like pointclass $\Gamma$ we can define the boldface inductive-like pointclass ${\bf \Gamma} = \bigcup\_{x \in \mathbb{R}} \Gamma(x)$ as usual (this does not have much to do with the particulars of "inductive-like.")


My question is, can we do the reverse? That is, given a boldface inductive-like pointclass ${\bf \Gamma}$ can we find a lightface inductive-like pointclass $\Gamma$ such that ${\bf \Gamma} = \bigcup\_{x \in \mathbb{R}} \Gamma(x)$? (We wouldn't expect uniqueness; if $\Gamma$ works then so does $\Gamma(z)$ for any real $z$.)


I have heard that Howard Becker proved something like this, but I can't find it and I don't know what kind of pointclasses it was for (probably more general than "inductive-like," which is merely the case I happen to be interested in at the moment.)


For a motivating example, consider the case where $\mathsf{AD}$ holds and $\bf \Gamma$ is the pointclass of $\kappa$-Suslin sets. If $\bf \Gamma$ is closed under $\forall^\mathbb{R}$ then it is boldface inductive-like, but as far as I know there is no canonical choice of lightface pointclass that corresponds to it.



Obtaining a lightface pointclass from a boldface one
See Lemma 3.4 and preceding paragraph of my paper "A characterization of jump operators", JSL vol. 53, 1988.


Howard Becker


