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
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) = 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,) =


if α = ,
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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