32,935 research outputs found
Half of an inseparable pair
A classical theorem of Luzin is that the separation principle holds for the
Pi^0_alpha sets but fails for the Sigma^0_alpha sets. We show that for every
Sigma^0_alpha set A which is not Pi^0_alpha there exists a Sigma^0_alpha set B
which is disjoint from A but cannot be separated from A by a Delta^0_alpha set
C. Assuming Pi^1_1-determancy it follows from a theorem of Steel that a similar
result holds for Pi^1_1 sets. On the other hand assuming V=L there is a proper
Pi^1_1 set which is not half of a Borel inseparable pair. These results answer
questions raised by F.Dashiell.
Latest version at: www.math.wisc.edu/~miller/Comment: LaTex2e 16 page
A Mad Q-set
A MAD (maximal almost disjoint) family is an infinite subset A of the
infinite subsets of {0,1,2,..} such that any two elements of A intersect in a
finite set and every infinite subset of {0.1.2...} meets some element of
in an infinite set. A Q-set is an uncountable set of reals such that every
subset is a relative G-delta set. It is shown that it is relatively consistent
with ZFC that there exists a MAD family which is also a Q-set in the topology
in inherits a subset of the Power set of {0,1,2,..}, ie the Cantor set.Comment: 13 pages, LaTeX2
The cardinal characteristic for relative gamma-sets
For a separable metric space define \pp(X) to be the smallest
cardinality of a subset of which is not a relative \ga-set in ,
i.e., there exists an \om-cover of with no \ga-subcover of . We give
a characterization of \pp(2^\om) and \pp(\om^\om) in terms of definable
free filters on \om which is related to the psuedointersection number \pp.
We show that for every uncountable standard analytic space that either
\pp(X)=\pp(2^\om) or \pp(X)=\pp(\om^\om). We show that both of following
statements are each relatively consistent with ZFC:
(a) \pp=\pp(\om^\om) < \pp(2^\om) and
(b) $\pp < \pp(\om^\om) =\pp(2^\om)
Ultrafilters with property (s)
A set X which is a subset of the Cantor set has property (s) (Marczewski
(Spzilrajn)) iff for every perfect set P there exists a perfect set Q contained
in P such that Q is a subset of X or Q is disjoint from X. Suppose U is a
nonprincipal ultrafilter on omega. It is not difficult to see that if U is
preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic
extension after forcing with the partial order of perfect sets, then U has
property (s) in the ground model. It is known that selective ultrafilters or
even P-points are preserved by Sacks forcing. On the other hand (answering a
question raised by Hrusak) we show that assuming CH (or more generally MA for
ctble posets) there exists an ultrafilter U with property (s) such that U does
not generate an ultrafilter in any extension which adds a new subset of omega.
http://www.math.wisc.edu/~miller/res/index.html [email protected]: LaTeX2e 10 page
Long Borel Hierarchies
We show that it is relatively consistent with ZF that the Borel hierarchy on
the reals has length . This implies that has countable
cofinality, so the axiom of choice fails very badly in our model. A similar
argument produces models of ZF in which the Borel hierarchy has length any
given limit ordinal less than , e.g., or
.
Latex2e: 24 pages plus 8 page appendix Latest version at:
www.math.wisc.edu/~mille
Souslin's Hypothesis and Convergence in Category
A sequence of functions f_n: X -> R from a Baire space X to the reals is said
to converge in category iff every subsequence has a subsequence which converges
on all but a meager set. We show that if there exists a Souslin Tree then there
exists a nonatomic Baire space X such that every sequence which converge in
category converges everywhere on a comeager set. This answers a question of
Wagner and Wilczynski, Convergence of sequences of measurable functions, Acta
Math Acad Sci Hung 36(1980), 125-128
Measurable rectangles
We give an example of a measurable set of reals E such that the set
E'={(x,y): x+y in E} is not in the sigma-algebra generated by the rectangles
with measurable sides. We also prove a stronger result that there exists an
analytic set E such that E' is not in the sigma-algebra generated by rectangles
whose horizontal side is measurable and vertical side is arbitrary. The same
results are true when measurable is replaced with property of Baire
On relatively analytic and Borel subsets
Define z to be the smallest cardinality of a function f:X->Y with X and Y
sets of reals such that there is no Borel function g extending f. In this paper
we prove that it is relatively consistent with ZFC to have b<z where b is, as
usual, smallest cardinality of an unbounded family in w^w. This answers a
question raised by Zapletal. We also show that it is relatively consistent with
ZFC that there exists a set of reals X such that the Borel order of X is
bounded but there exists a relatively analytic subset of X which is not
relatively coanalytic. This answers a question of Mauldin.Comment: LaTeX2e 10 pages available at
http://www.math.wisc.edu/~miller/res/index.htm
On squares of spaces and Fsigma-sets
We show that the continuum hypothesis implies there exists a Lindelof space X
such that X x X is the union of two metrizable subspaces but X is not
metrizable. This gives a consistent solution to a problem of Balogh, Gruenhage,
and Tkachuk. The main lemma is that assuming the continuum hypothesis there
exist disjoint sets of reals X and Y such that X is Borel concentrated on Y,
(i.e., for any Borel set B if Y is contained in B then X-B is countable,) but
(X x X - diagonal) is relatively Fsigma in (X x X) U (Y x Y).Comment: 6 page
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