# Descriptive set theory on $\mathbb{R}^\mathbb{N}$

The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology on each $\mathbb{R}$-factor? (Allowing, for the moment, the use of the term "descriptive set theory" in this context; since $\mathbb{R}^\mathbb{N}$ is not even second-countable under the given topology, it is certainly not a Polish space, so maybe the term "descriptive set theory" isn't quite right here.)

Here's my motivation. I've learned a small amount of descriptive set theory on the Baire space $\mathbb{N}^\mathbb{N}$, and separately I've learned the proof of full (quasi-)Borel determinacy. The latter theorem is a statement about arbitrary spaces of the form $X^\mathbb{N}$ with $X$ given the discrete topology and $X^\mathbb{N}$ given the product topology, so I became interested in what one can say about such spaces with $X$ uncountable. For example, some determinacy statements which are consistent (relative to large cardinals) for $\mathbb{N}^\mathbb{N}$ are inconsistent when applied to $\mathbb{R}^\mathbb{N}$. But even though the topology $\mathbb{R}^\mathbb{N}$ is in many ways "bad," it is also not totally unnatural; besides arising when thinking about Baire space, I've also run into it when playing around with various notions of forcing.

So my question is, what is good source to learn about this topology on $\mathbb{R}^\mathbb{N}$? Specifically, I'm interested in results with descriptive set theoretic flavor, e.g., results about pointclasses analogous to the various projective pointclasses. Hopefully this question is not too vague.

-
I believe A. H. Stone did some work on this. Maybe try "Non-Separable Borel Sets" I and II. – Gerald Edgar Dec 8 '12 at 21:23