Nomenclature used to refer to the power set $\mathcal{P}$ as a pointclass

Question

This may be silly and vague question, but I'm writing a presentation on the Wadge hierarchy were I refer to different boldface pointclasses , like $\boldsymbol{\Delta}_1^1(X)$ (Borel sets) or $\text{Proj}(X)$ (projective sets), and in the discourse I need to refer to the power set of a space (of its domain) $\mathcal{P}(X)$ as a boldface pointclass. The reason why is that in this presentation I want to develop, in parallel, the theory within these three different pointclasses, therefore I find myself in refering to $\mathcal{P}(X)$ as a pointclass, for homogeneity reasons.

Is this legit? I mean, it should be, but I'm asking if it isn't a too of a stretch.
Is there a standard notation for it, like "discrete" pointclass?

Thanks

Answer 1

(I'm turning an extended version of my comment above into an answer, in order to close the issue.)

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I haven't seen any specific name for the powerset as an “all encompassing” pointclass; note that all usual pointclasses refer to definable subsets of the relevant spaces. The definition adopted by Moschovakis in his book does not explicitely preclude $\mathcal{P}$ as a pointclass, but he states

In definability theory we typically start with a small pointclass $\Lambda$ and certain operations on pointsets and then we study the sets which can be constructed by applying (once or repeatedly) the given operations to the members of $\Lambda$.

In summary, there is nothing incorrect in saying that the powerset is a pointclass, but you will probably want to say that it isn't “morally” one (unless, for instance, you live inside the constructible universe).