Lecture Notes on Descriptive Set Theory
Jan Reimann
Department of Mathematics
Pennsylvania State University
Notation
U (x)
U
2<
, , . . .
2
n
[]
Ball of radius about x
Topological closure of U
Set of nite binary strings
Finite binary strings
Cantor space,
References
M. Gromov. Metric structures for Riemannian and non-Riemannian spaces, volume
152 of Progress in Mathematics. Birkhuser Boston Inc., Boston, MA, 1999.
P R. Halmos. Measure Theory. Van Nostrand, 1950.
.
T. Jech. Set Theory. Springer Monographs i
Lecture 10: The Structure of Borel Sets
In this lecture we further investigate the structure of Borel sets. We will use
the results of the previous lecture to derive various closure properties and other
structural results. As an application, we see that t
Lecture 13: Regularity Properties of Analytic Sets
In this lecture we verify that the analytic sets are Lebesgue measurable (LM)
and have the Baire property (BP). Since both properties are closed under complements, they also hold for the class of co-analy
Lecture 18: 1 Sets
2
In this lecture we extend the results of the previous lecture to 1 sets.
2
Tree representations of 1 sets
2
Analytic sets are projections of closed sets. Closed sets are in are innite
paths through trees on .
We call a set A Y -Sousli
Lecture 19: Recursive Ordinals and Ordinal Notations
We have seen that the property codes a well-ordering of is important for
the study of co-analytic sets. If A is 1 , then there exists a tree T on such
1
that
A T () is well-founded.
If T () is well-fou
Lecture 14: The Projective Hierarchy
In Lecture 12 we saw that the analytic sets are not closed under complements,
which led us to the introduction of the co-analytic sets as a separate class.
We saw analytic sets are projections of closed sets and hence
Lecture 6: Measure and Category
The Borel hierarchy classies subsets of the reals by their topological complexity.
Another approach is to classify them by size.
Filters and Ideals
The most common measure of size is, of course, cardinality. In the presence
Lecture 5: Borel Sets
Topologically, the Borel sets in a topological space are the -algebra generated by
the open sets. One can build up the Borel sets from the open sets by iterating the
operations of complementation and taking countable unions. This gen
Lecture 4: Trees
Let A be a set. The set of all nite sequences over A is denoted by A< .
Denition 4.1: A tree on A is a set T A< that is closed under prexes, that is
, [ T & T ]
We call the elements of T nodes.
A sequence A is a innite path through or inn
Lecture 3: Excursion The Urysohn Space
Recall that a mapping e : X Y between two metric spaces (X , dX ) and (Y, dY )
is an isometry if
dY ( f (x), f ( y) = dX (x, y)
for all x, y X ,
that is, an isometry is a mapping that preserves distances. f is also c
Lecture 2: Polish Spaces
The proofs in the previous lecture are quite general, that is, they make little use
of specic properties of . If we scan the arguments carefully, we see that we
can replace by any metric space that is complete and contains a count
Lecture 11: Continuous Images of Borel Sets
In 1916, Nikolai Lusin asked his student Mikhail Souslin to study a paper by
Henri Lebesgue. Souslin found a number of errors, including a lemma that
asserted that the projection of a Borel is again Borel. In th
Lecture 9: Effective Borel sets
Suppose U is open. The there exists a set W < such that
U=
N .
W
Using a standard (effective) coding procedure, we can identify nite sequence
of natural numbers with a natural number, and thus can see W as a subset of .
If
Lecture 1: Perfect Subsets of the Real Line
Descriptive set theory nowadays is understood as the study of denable subsets
of Polish Spaces. Many of its problems and techniques arose out of efforts to
answer basic questions about the real numbers. A promin
Lecture 21: Co-analytic Ranks
In the previous lecture we learned about how 1 set can be analyzed in terms of
1
countable ordinals. In this lecture we will deepen this analysis. We will develop
the theory of 1 -ranks, which is a powerful tool in descriptiv
Lecture 20: 1 Sets of Natural Numbers
1
In this lecture we consider 1 sets of natural numbers. They are dened just
1
like their counterparts in . Using the Kleene Normal Form, a set X is 1
1
if there exists a bounded formula (x, y, , ) such that
xX
y (x,
Lecture 7: Measure and Category
The Borel hierarchy classies subsets of the reals by their topological complexity.
Another approach is to classify them by size.
Filters and Ideals
The most common measure of size is, of course, cardinality. In the presence
Lecture 22: Hyperarithmetical Sets
Is there an effective counterpart to Souslins Theorem that Borel = 1 ? Den1
ability in second order arithmetic gives us the lightface classes 0 and 0 for
n
n
nite n, but what would a lightface 0 set be?
Instead of denabi
Calculus 1, Chapter 4 Study Guide
Prepared by Dr. Robert Gardner
The following is a brief list of topics covered in Chapter 4 of Thomas
Calculus. Test questions will be chosen directly from the text. This list is
not meant to be comprehensive, but only gi
Lecture 16: Constructible Reals
In this lecture we transfer the results about L to the projective hierarchy. The
main idea is to relate sets of reals that are dened by set theoretic formulas to
sets dened in second order arithmetic.
The effective projecti
Lecture 15: The Constructible Universe
A set X is rst-order denable in a set Y (from parameters) if there exists a rstorder formula (x 0 , x 1 , . . . , x n ) in the language of set theory (i.e. only using the
binary relation symbol ) such that for some a
Lecture 17: Co-Analytic Sets
In the previous lecture we saw how to translate set theoretic denitions of sets
of reals into second order arithmetic. One can ask the converse question does
denability in second order arithmetic imply constructibility? We wil
Lecture 6: Borel Sets as Clopen Sets
In this lecture we will learn that the Borel sets have the perfect subset property,
which we already saw holds for closed subsets of Polish spaces.
The proof changes the underlying topology so that all Borel sets becom
Lecture 8: The Axiom of Choice
In the previous lectures, a number of regularity principles for sets of real
numbers emerged.
(PS) The perfect subset property,
(LM) Lebesgue measurability,
(BP) the Baire property.
We have seen that the Borel sets in have a