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1
BOREL AND BAIRE REDUCIBILITY
by
Harvey M. Friedman
Department of Mathematics
Ohio State University
Columbus, Ohio 43210
friedman@math.ohiostate.edu
www.math.ohiostate.edu/~friedman/
May 25, 1999
INTRODUCTION
The Borel reducibility theory of Polish equivalence
relations, at least in its present form, was initiated in
[FS89]. There is now an extensive literature on this topic,
including fundamental work on the GlimmEffros dichotomy in
[HKL90], on countable Borel equivalence relations in [DJK94],
and on Polish group actions in [BK96]. Current principal
contributors include H. Becker, R. Dougherty, L. Harrington,
G. Hjorth, S. Jackson, A.S. Kechris, and A. Louveau, R. Sami,
and S. Solecki.
A Polish space is a topological space that is separable and
completely metrizable. The Borel subsets of a Polish space
form the least
σ
algebra containing the open subsets. A Borel
function from one Polish space to another is a function such
that the inverse image of every open set is Borel. Two Polish
spaces are Borel isomorphic if and only if there is a oneone
onto Borel function from the first onto the second. This is
an equivalence relation. Any two uncountable Polish spaces
are Borel isomorphic. See [Ke94].
We also consider Baire measurable subsets of a Polish space.
A nowhere dense set in a Polish space is a set whose closure
contains no nonempty open set. A meager subset of a Polish
space is a set which is the countable union of nowhere dense
sets. A Baire (measurable) subset of a Polish space is a set
whose symmetric difference with some open set is meager. A
comeager subset of a Polish space is a subset whose
complement is meager. We have the fundamental Baire category
theorem: A Polish space is not meager.
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A function from one Polish space to another is said to be
Baire if and only if the inverse image of every open set is
Baire.
A Polish equivalence relation is a pair (X,E), where X is a
Polish space and E is an equivalence relation on X. A Borel
(analytic) equivalence relation is a Polish equivalence
relation (X,E), where E is as a Borel measurable (analytic)
subset of X
2
.
Let (X,E
1
) and (Y,E
2
) be Polish equivalence relations. We say
that h:X
→
Y is a reduction from (X,E
1
) into (Y,E
2
) if and
only if for all x,y
∈
X, E
1
(x,y)
↔
E
2
(h(x),h(y)). We say that
(X,E1) is Borel reducible to (Y,E
2
) if and only if there is a
Borel reduction from (X,E
1
) to (Y,E
2
). We also say that (X,E
1
)
is Baire reducible to (Y,E
2
) if and only if there is a Baire
reduction from (X,E
1
) to (Y,E
2
).
Let STR = (STR,
≈
) be the analytic equivalence relation of all
structures of countable similarity type (in the sense of
model theory) whose domain is
ω
, under isomorphism. Also let
BR = (P(
ω
x
ω
),
≈
), be the analytic equivalence relation of
binary relations on
ω
under isomorphism. It is well known
that BR is not a Borel equivalence relation and BR is Borel
isomorphic to STR. It is generally simpler to work with BR.
The condition that a Polish equivalence relation E is Borel
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