Confusion About Pointclasses

Question

I am doing some work learning about the Axiom of Determinacy and its consequences. This has led me to learning about the properties of the Baire space, $\omega^\omega$. I have recently come across the concept of pointclasses, given with the following definition. $\Gamma$ is a pointclass if it consists of pairs $(A,X)$, where A is a subset of the Polish space $X$. I am having trouble understanding the definition and figuring out exactly what a point class is. Are there any examples of pointclasses that would be helpful to keep in mind? Or is there an intuitive explanation of exactly what a pointclass is?

Answer 1

The standard examples are:

• The open sets. This is the pointclass consisting of pairs $(A, X)$ where $X$ is a Polish space and $A\subseteq X$ is open.

• The closed sets. This is the pointclass consisting of pairs $(A, X)$ where $X$ is a Polish space and $A\subseteq X$ is closed.

Similarly, we also have:

• The $G_\delta$ sets.

• The Borel sets.

• The analytic sets. (That is, pairs $(A, X)$ where $X$ is a Polish space and $A\subseteq X$ is the continuous image of some Borel subset of $X$.)

• The coanalytic sets.

Etc.

We also have silly examples, like:

• The set of pairs $(A, X)$ where $X$ is a Polish space and $A$ has fewer than 17 points.

• The set containing just $(\{\pi\},\mathbb{R})$.

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Basically, think of a pointclass as the set of all pointsets which are describable in some nice way. For instance, the Borel pointclass is the pointclass of all sets which have Borel codes; the open pointclass is the pointclass of all sets which are formed by taking unions of open balls; and so forth.

In descriptive set theory, you'll be interested in pointclasses that satisfy reasonable properties; for instance, being closed under continuous preimages. These closure properties are what the silly examples described above lack.