CHAPTER 1

THE BASIC CLASSICAL NOTIONS

Let = {0,1,2,...} be the set of (nonnegative) integers and let R be the set of real

numbers. The main business of Descriptive Set Theory is the study of , R and their

subsets, with particular emphasis on the definable sets of integers and reals. Another

fair name for it is Definability Theory for the Continuum.

In this first chapter we will introduce some of the basic notions of the subject and

we will establish the elementary facts about them.

1A. Perfect Polish spaces

Instead of working specifically with the reals, we will frame our results in the wider

context of complete, separable metric spaces (Polish spaces) with no isolated points

(perfect). One of the reasons for doing this is the wider applicability of the theory

thus developed. More than that, we often need to look at more complicated spaces in

order to prove results about R.(1–5)

Of course R is a perfect Polish space and so is the real n-space

Rn

for each n ≥ 2.

There are two other important examples of such spaces which will play a key role in

the sequel.

Baire space is the set of all infinite sequences of integers (natural numbers),

N =

with the natural product topology, taking discrete. The basic neighborhoodsare of

the form

N(k0,...,kn) = {α ∈ N : α(0) = k0,...,α(n) = kn},

one for each tuple k0,...,kn. We picture N as (the set of infinite branches of) a tree,

where each node splits into countably many one-point extensions, Figure 1A.1.

It is easy to verify that the topology of N is generated by the metric

d(α,) =

⎧

⎨0,

⎩

if α = ,

1

least n[α(n) = (n)]+1

, if α = .

Also, N is complete with this metric and the set of ultimately constant sequences is

countable and dense in N,so N is a perfect Polish space.

One can show that N is homeomorphic with the set of irrational numbers, topol-

ogized as a subspace of R. The proof appeals to some basic properties of continued

fractions and does notconcernus here—itcan be found in any good bookon number

theory, for example Hardy and Wright [1960]. Although we will never use this result,

we will find it convenient to call the members of N irrationals.

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http://dx.doi.org/10.1090/surv/155/02