Abstracts for Special Session on
Modern Methods in Set Theory and Topology

Abstract: We will present an elementary submodel proof of Yiang Shouli's theorem that strict p-spaces are submetacompact.

Tomek Bartoszynski

Abstract: I will present some recent results concerning cardinal invariants, small sets of reals and their properties.

Andreas Blass

Abstract: I shall discuss
(1) some cardinal characteristics associated with ultrafilters on the natural numbers,
(2) some connections between these and more familiar cardinal characteristics of the continuum, and
(3) some consequences that involve only those more familiar characteristics.

Ilijas Farah

Abstract: Long ago, S. Ulam has suggested the study of what he called the stability of the solutions to certain functional equations, namely the question whether an approximate solution is always in some sense close to a solution of the given equation. I shall discuss this question in case of embeddings of Boolean algebras, groups and semilattices, and its connection to the existence of pathological submeasures.

William Fleissner

Abstract: A natural way to prove that metric spaces are D-spaces is "well-order, with large neighborhoods first." It is not hard to generalize to show that semistratifiable spaces are D-spaces. Looking carefully at the proof, we abstract the weakest sufficient hypothesis, and then prove that the hypothesis is equivalent to semistratifiability.

Gary Gruenhage

Abstract: We survey results related to Husek's still open problem of whether or not a compact space with a small diagonal must be metrizable, and show that for countably compact spaces a positive answer is consistent with and independent of ZFC.

Claude Laflamme

Abstract: We present a selective overview of filters (on the naturtal numbers) as a Set Theoretical and Combinatorial tool to attack various problems in algebra, analysis and topology.

Justin Moore

Abstract: A coloring $c:[X]^n \to Z$ is said to be irreducible if for every $Y \subseteq X$ of equal cardinality $c'' [Y]^n = Z$. The focus of this talk will be to show that there are continuous irreducible colorings on sets of reals associated with various cardinal invariants of the continuum. It is interesting that some of the colorings make crucial use of exponential lower bounds which have been proven for a certain class of finite Ramsey numbers.

Peter Nyikos

Abstract: A manifold is a connected Hausdorff space in which every point has a neighborhood homeomorphic to Euclidean n-space (n is unique). A space is {\it collectionwise Hausdorff (cwH)} if every closed discrete subspace $D$ can be expanded to a disjoint collection of open sets each of which meets $D$ in one point. There are exactly two examples of 1-dimensional nonmetrizable hereditarily normal, hereditarily cwH manifolds: the long line and the long ray. On the other hand: {\bf Theorem.} If it is consistent that there is a supercompact cardinal, it is consistent that every every hereditarily normal, hereditarily cwH manifold of dimension greater than 1 is metrizable. In contrast, the Rudin-Zenor 2-manifold constructed using the Continuum Hypothesis is perfectly normal (hence hereditarily normal and hereditarily cwH) and nonmetrizable.

Judith Roitman

Abstract: Let $X$ be countable. Which partial orders on $X$ extend to linear orders on $X$ isomorphic to ${\Bbb Q}$? A necessary and sufficient condition is found which easily implies previously known partial results (either necessary or sufficient) plus some new ones.

Andrzej Roslanowski

Abstract: We will present a survey of three recent joint papers with Saharon Shelah that concern cardinal invariants of Boolean algebras. We will show how some topological cardinal functions, if viewed from the point of view of Boolean algebras, give rise to "finite" (or: "bounded") versions of the invariants. Furthermore we will point out some forcing constructions which can be done for both topological and Boolean algebraic versions of some cardinal functions, but which seem to be easier to deal with in the second case.

Marion Scheepers

Abstract: After a reminder of the rules of the point-picking game studied by Berner and Juhasz during the early 1980's, we give a brief survey of their results. Then we describe how their game is connected with the combinatorics of open covers. Then we describe a new example of one of the phenomena discovered by Berner and Juhasz and suggest a promising attack on one of their open problems.

John J. Schommer (joint work with Mary Anne Swardson)

Abstract: One of the standard definitions of almost realcompact is the following: every ultrafilter of open sets with the closed countable intersection property is fixed. If we replace the open sets of this definition with cozero-sets, we get yet another generalization of realcompact, one we'll call almost$^*$ realcompact. Perhaps surprisingly, almost and almost$^*$ realcompact are independent of each other. In this talk we will look at the relevant counter-examples, as well as sketch proofs of the following ``cures" to that independence: {\bf Proposition.} If $X$ is almost$^*$ realcompact and almost weak Oz, then $X$ is almost realcompact. {\bf Proposition.} If $X$ is almost realcompact and strongly countably paracompact, then $X$ is almost$^*$ realcompact.

Adrienne Stanley

Abstract: We will discuss normal subspaces of finite products of ordinals and how they relate to collectionwise normality and other topological properties.

Franklin D. Tall

Abstract: We discuss recent work of the author and his students concerning the topology of elementary submodels.

Jerry E. Vaughan (joint work with Richard E. Hodel)

Abstract: Recall that $\phi$ is a {\em cardinal function} provided $\phi$ is a function from the class of all topological spaces into the class of all infinite cardinal numbers such that if $X$ and $Y$ are homeomorphic then $\phi(X) = \phi(Y)$. We discuss relations among the following three properties which arise in the study of reflection theorems for cardinal functions. (1) A cardinal function $\phi$ {\em reflects} a cardinal $\kappa$ in a class of topological spaces ${\cal C}$ means: for every $X\in {\cal C}$, if $\phi(X) \geq \kappa$ then there exists $Y\in[X]^{\leq\kappa}$ such that $\phi(Y) \geq\kappa$. (2) A cardinal function $\phi$ {\em strongly reflects} a cardinal $\kappa$ in a class of topological spaces ${\cal C}$ means: for every $X\in {\cal C}$, if $\phi(X) \geq \kappa$ then there exists $Y\in[X]^{\leq\kappa}$ such that $\phi(Z) \geq\kappa$ for all $Y\subset Z\subset X$. (3) A cardinal function $\phi$ satisfies {\em $IU(\kappa)$} in a class of topological spaces ${\cal C}$ means: for every $X\in {\cal C}$, if $X = \cup\{X_\alpha: \alpha < \lambda\}$ is an increasing union (i.e., $\alpha < \beta$ implies $X_\alpha \subset X_\beta$) with $\kappa < \lambda$, $\lambda$ regular, and $\phi(X_\alpha) < \kappa$ for all $\alpha < \lambda$, then $\phi(X) < \kappa$.

Stephen Watson (joint work with Kerry Richardson)

Abstract: Resolutions were introduced by V. V. Fedorcuk around 1970 as a method for constructing counterexamples in the dimension theory of compacta. In 1992, we developed resolutions as a far-reaching method of constructing topological spaces. In 1994, we developed a wider theory of resolution by multifunctions which in some cases is equivalent to fully closed. In this article, we report on an even wider notion of resolution which turns out to be equivalent to a natural property of continuous mappings and also have a simple categorical characterization. We give some applications of this method. This work and that of 1994 will appear in a joint paper with Kerry Richardson.

Kohzo Yamada

Abstract: It is known that if a space $X$ is non-discrete, then neither the free topological group $F(X)$ nor the free abelian topological group $A(X)$ is first countable. We give equivalent conditions of a metrizable space $X$ such that the subspaces $F_n(X)$ of $F(X)$ and the subspaces $A_n(X)$ of $A(X)$ are metrizable for each natural number $n$, respectively. In addition, we prove that if $F_n(X)$ ($A_n(X)$) is first countable, then it becomes metrizable for each $n\geq2$.