Justin Tatch Moore

A solution to the L space problem

In this paper I will construct a non-separable hereditarily Lindelof space (L space) without any additional axiomatic assumptions. I will also show that there is a function f from [omega_1]^2 to omega_1 such that if A,B, subsets of omega_1, are uncountable and x omega_1, then there are a < b in A and B respectively with f(a,b) = x. Previously it was unknown whether such a function existed even if omega_1 was replaced by 2. Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality aleph_1. Each of these results gives a strong refutation of a well known and longstanding conjecture. The results all stem from the analysis of oscillations of coherent sequences {e_i : i < omega_1} of finite-to-one functions. I expect that the methods presented will have other applications as well.

SET MAPPING REFLECTION

In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2ω = ω2 and that satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that □(κ) fails for all regular κ > ω1.

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, ¥o1, ¥o.1, C, C. where X is any suborder of the reals of cardinality .1 and C is any Countryman line. This confirms a longstanding conjecture of Shelah.

The Proper Forcing Axiom

The Proper Forcing Axiom is a powerful extension of the Baire Category Theo- rem which has proved highly effective in settling mathematical statements which are independent of ZFC. In contrast to the Continuum Hypothesis, it eliminates a large number of the pathological constructions which can be carried out using additional axioms of set theory.

Some of the combinatorics related to Michael’s problem

We present some new methods for constructing a Michael space, a regular Lindelöf space which has a non-Lindelöf product with the space of irrationals. The central result is a combinatorial statement about the irrationals which is a necessary and sufficient condition for the existence of a certain class of Michael spaces. We also show that there are Michael spaces assuming d = cov(M) and that it is consistent with cov(M) < b < d that there is a Michael space. The influence of Cohen reals on Michael’s problem is discussed as well. Finally, we present an example of a Michael space of weight less than b under the assumption that b = d = cov(M) = אω+1 (whose product with the irrationals is necessarily linearly Lindelöf).

Open colorings, the continuum and the second uncountable cardinal

The purpose of this article is to analyze the cardinality of the continuum using Ramsey theoretic statements about open colorings or open coloring axioms. In particular it will be shown that the conjunction of two well-known axioms, OCA [ARS] and OCA [T] , implies that the size of the continuum is N 2 .

PERFECT COMPACTA AND BASIS PROBLEMS IN TOPOLOGY

Publisher Summary This chapter discusses the concept of perfect compacta and some basis problems in topology. An interesting example of a compact Hausdorff space that is often presented is the unit square [0, 1]×[0, 1] with the lexicographic order topology. The closed subspace consisting of the top and bottom edges is perfectly normal. This subspace is often called the Alexandroff double arrow space. It is also sometimes called the “split interval”, because it can be obtained by splitting each point x of the unit interval into two points x0, x1, and defining an order by declaring x0 < x1 and using the induced order of the interval otherwise. The top edge of the double arrow space minus the last point is homeomorphic to the Sorgenfrey line, as is the bottom edge minus the first point. Hence it has no countable base, so being compact, is non-metrizable. There is an obvious two-to-one continuous map onto the interval. The concepts of uncountable spaces are also discussed in the chapter. A discussion on different approaches and axiomatics is also presented.

A Gδ ideal of compact sets strictly above the nowhere dense ideal in the Tukey order

Abstract We prove that there is a G δ σ -ideal of compact sets which is strictly above NWD in the Tukey order. Here NWD is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Velickovic asked in [Alain Louveau, Boban Velickovic, Analytic ideals and cofinal types, Ann. Pure Appl. Logic 99 (1–3) (1999) 171–195].

A nonamenable finitely presented group of piecewise projective homeomorphisms

In this article we will describe a finitely presented subgroup of the group of piecewise projective homeomorphisms of the real projective line. This in particular provides a new example of a finitely presented group which is nonamenable and yet does not contain a nonabelian free subgroup. It is in fact the first such example which is torsion free. We will also develop a means for representing the elements of the group by labeled tree diagrams in a manner which closely parallels Richard Thompson’s group F .

Fast growth in the Følner function for Thompson’s group

The purpose of this note is to prove a lower bound on the growth of Folner functions for Thompson’s group F . Specifically I will prove that, for any finite generating set Γ ⊆ F , there is a constant C such that FolF,Γ(C) ≥ expn(0).