1
SELECTION FOR BOREL RELATIONS
by
Harvey M. Friedman*
Department of Mathematics
Ohio State University
[email protected]
http://www.math.ohiostate.edu/~friedman/
September 30, 2001
Abstract. We present several selection theorems for Borel
relations, involving only Borel sets and functions, all of
which can be obtained as consequences of closely related
theorems proved in [DSR 96,99,01,01X] involving coanalytic
sets. The relevant proofs given there use substantial set
theoretic methods, which were also shown to be necessary. We
show that none of our Borel consequences can be proved
without substantial set theoretic methods. The results are
established for Baire space. We give equivalents of some of
the main results for the reals.
Introduction.
Let S be a set of ordered pairs and A be a set. We say that f
is a selection for S on A if and only if dom(f) = A and for
all x A, (x,f(x)) S. We say that f is a selection for S
if and only if f is a selection for S on {x: (
$
y)((x,y)
S)}.
Let N be the set of all nonnegative integers. 2
N
= {0,1}
N
is
Cantor space, where {0,1} is given the discrete topology. N
N
is Baire space, where N is given the discrete topology.
We use for the reals with the usual topology. All results
in sections 15 are formulated on N
N
. This is most convenient
for the proofs. In section 6 we give equivalent formulations
on of some of the main results.
The following result from [DSR01X] led to this research.
PROPOSITION I. Let S
Õ
N
N
x N
N
be coanalytic and E
Õ
N
N
be
Borel. If there is a continuous selection for S on every
compact subset of E, then there is a continuous selection for
S on E.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
Proposition I is proved in [DSR01X] using set theoretic
assumptions going beyond ZFC. In fact, [DSR01X] shows that
Proposition I is provably equivalent to
COUNT. For all f N
N
, N
N
L[f] is countable.
over ZFC.
[DSR01X] also gives the comparatively simple proof of
Proposition I using analytic determinacy, which works for all
coanalytic E. Moreoever, [DSR01X] shows that Proposition I
for coanalytic E is equivalent to analytic determinacy.
The proof of Proposition I from COUNT in [DSR01X] is rather
complicated, and we conjecture that the proof can be
considerably simplified using additional methods from modern
set theory.
In section 1, we present the simple [DSR01X] proof of
Proposition I from analytic determinacy. It is obvious that
if “coanalytic” is replaced by “Borel”, then the proof goes
through unmodified with Borel determinacy instead of analytic
determinacy. Therefore we have the following theorem of ZFC.
THEOREM II. Let S
Õ
N
N
x N
N
be Borel and E
Õ
N
N
be Borel. If
there is a continuous selection for S on every compact subset
of E, then there is a continuous selection for S on E.
In section 4, we show that Theorem II cannot be proved using
only countably many iterations of the power set operation.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 JOSHUA
 Math, Topology, Natural number, Borel, Borel set, Descriptive set theory

Click to edit the document details