77 research outputs found
Covering an uncountable square by countably many continuous functions
We prove that there exists a countable family of continuous real functions
whose graphs together with their inverses cover an uncountable square, i.e. a
set of the form , where is an uncountable subset of the real
line. This extends Sierpi\'nski's theorem from 1919, saying that
can be covered by countably many graphs of functions and inverses of functions
if and only if the size of does not exceed . Our result is also
motivated by Shelah's study of planar Borel sets without perfect rectangles.Comment: Added new results (9 pages
A five element basis for the uncountable linear orders
In this paper I will show that it is relatively consistent with the usual
axioms of mathematics (ZFC) together with a strong form of the axiom of
infinity (the existence of a supercompact cardinal) that the class of
uncountable linear orders has a five element basis. In fact such a basis
follows from the Proper Forcing Axiom, a strong form of the Baire Category
Theorem. The elements are X, omega_1, omega_1^*, C, C^* where X is any suborder
of the reals of cardinality aleph_1 and C is any Countryman line. This confirms
a longstanding conjecture of Shelah.Comment: 21 page
The embedding structure for linearly ordered topological spaces
In this paper, the class of all linearly ordered topological spaces (LOTS)
quasi-ordered by the embeddability relation is investigated. In ZFC it is
proved that for countable LOTS this quasi-order has both a maximal (universal)
element and a finite basis. For the class of uncountable LOTS of cardinality
it is proved that this quasi-order has no maximal element for
at least the size of the continuum and that in fact the dominating number for
such quasi-orders is maximal, i.e. . Certain subclasses of LOTS, such
as the separable LOTS, are studied with respect to the top and internal
structure of their respective embedding quasi-order. The basis problem for
uncountable LOTS is also considered; assuming the Proper Forcing Axiom there is
an eleven element basis for the class of uncountable LOTS and a six element
basis for the class of dense uncountable LOTS in which all points have
countable cofinality and coinitiality
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
Algebraic properties of profinite groups
Recently there has been a lot of research and progress in profinite groups.
We survey some of the new results and discuss open problems. A central theme is
decompositions of finite groups into bounded products of subsets of various
kinds which give rise to algebraic properties of topological groups.Comment: This version has some references update
Characterization and Transformation of Countryman Lines and R-Embeddable Coherent Trees in ZFC
Ph.DDOCTOR OF PHILOSOPH
Category forcings, , and generic absoluteness for the theory of strong forcing axioms
We introduce a category whose objects are stationary set preserving complete
boolean algebras and whose arrows are complete homomorphisms with a stationary
set preserving quotient. We show that the cut of this category at a rank
initial segment of the universe of height a super compact which is a limit of
super compact cardinals is a stationary set preserving partial order which
forces and collapses its size to become the second uncountable
cardinal. Next we argue that any of the known methods to produce a model of
collapsing a superhuge cardinal to become the second uncountable
cardinal produces a model in which the cutoff of the category of stationary set
preserving forcings at any rank initial segment of the universe of large enough
height is forcing equivalent to a presaturated tower of normal filters. We let
denote this statement and we prove that the theory of
with parameters in is generically invariant
for stationary set preserving forcings that preserve . Finally we
argue that the work of Larson and Asper\'o shows that this is a next to optimal
generalization to the Chang model of Woodin's generic
absoluteness results for the Chang model . It remains open
whether and are equivalent axioms modulo large cardinals
and whether suffices to prove the same generic absoluteness results
for the Chang model .Comment: - to appear on the Journal of the American Mathemtical Societ
Cycle decompositions: from graphs to continua
We generalise a fundamental graph-theoretical fact, stating that every
element of the cycle space of a graph is a sum of edge-disjoint cycles, to
arbitrary continua. To achieve this we replace graph cycles by topological
circles, and replace the cycle space of a graph by a new homology group for
continua which is a quotient of the first singular homology group . This
homology seems to be particularly apt for studying spaces with infinitely
generated , e.g. infinite graphs or fractals.Comment: Advances in Mathematics (2011
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