Kunen Fest : Topology and Set Theory Conference

Box products: then and later

A survey of results relating to the question "Which box products are paracompact?", especially work which relates to or stems from Kunen's insights into the problem.

In early 2008 S. Shelah published a proof of diamond for the successors of uncountable cardinals witnessing the localised GCH. We look at the various ways the methods in this proof can be applied (or fail to be applied) to other diamond-like guessing sequences on cardinals above omega_1.

Lelek's problem is not a metric problem

Lelek defined the span of metric continua and asked whether a continuum with span zero must be chainable. This question makes sense also in the non-metric setting as having span zero does not depend on the metric but rather on whether certain subcontinua of the square of the space meet the diagonal or not. We show that a non-metric counterexample can be converted into a metric one. Joint work with: Dana Bartosova and Berd van der Steeg.

Infinite dimensional perfect set theorems

What largeness condition on an analytic set $A \subseteq [0,1]^{\omega}$ can guarantee that there is a nonempty perfect $P \subseteq [0,1]$ satisfying $P^{\omega} \subseteq A$? We obtain infinite dimensional analogues of some classical perfect set theorems, and present surprising counterexamples.

Towards a rough structure theory of Borel graph homomorphisms

$\pi$-character and the order types of local bases in compacta

Given a compactum X in which every local base is well quasi-ordered with respect to containment, some point has countable $\pi$-character. This generalizes the much easier proposition that compacta with well-ordered local bases at all points are have points with countable character.

For every known homogeneous compactum with character $\kappa$, every local base has a subset of $\kappa$-many elements pairwise incomparable with respect to inclusion. This is not true of inhomogeneous compacta.

An inevitable extension of ZFC

Let ZFC* be the recursive extension of ZFC obtained by adding a scheme that stipulates the existence of n-Mahlo cardinals that are Sigma_n-reflective for each standard natural number n. In this talk I will discuss some old and new results indicating that ZFC* is the "right" extension of ZFC.

Loops, quasigroups and automated reasoning

In the last dozen or so years, research in loop and quasigroup theory has been dominated by the use of automated deduction tools. Ken Kunen is almost entirely responsible for this paradigm shift within the field. In this talk, I will survey Ken's work in algebra, focusing primarily on his results in loop and quasigroup theory, and then I will discuss more recent developments. Since it is expected that the bulk of the audience share Ken's other interests but not his algebraic ones, this talk will be aimed at a general mathematical audience. Expect an anecdote or two.

Maharam's problem was first posed in 1947 and it remained unsolved until 2006 when it was finally settled by M. Talagrand. We introduce Maharam's problem and outline Talagrand's perplexing solution.

Locally Connected HS/HL Compacta

We consider a class of compacta $X$ such that the maps from $X$ onto metric compacta define an Aronszajn tree of closed subsets of $X$. Under $\diamondsuit$, there are Aronszajn compacta that are both hereditarily separable (HS) and hereditarily Lindel\"of (HL), and can be constructed to be zero-dimensional or locally connected.

We also look at the open question: Is there (in ZFC) a compact space which is non-metrizable, HS, HL, and locally connected?

Lindel\"of spaces, D-spaces, and selection principles

abstract in PDF format

We report on recent research in collaboration with Marion Scheepers and with Leandro Aurichi. Classical combinatorial strengthenings of Lindel\"ofness, namely the Menger and Rothberger properties, yield new insights into longstanding open problems in topology. For example,

Theorem 1. If it is consistent that there is a supercompact cardinal, it is consistent with GCH that all Rothberger spaces with points $G_\delta$ have cardinality $\leq \aleph_1$, and that all uncountable Rothberger spaces of character $\leq \aleph_1$ have Rothberger subspaces of size $\aleph_1$.

Theorem 2. Every Rothberger space with points $G_\delta$ has cardinality less than the first real-valued measurable cardinal.

Theorem 3. Menger spaces are $D$-spaces.

Theorem 4. CH implies that if a $T_3$ space $X$ is either separable or first countable, and if $X \times Y$ is Lindel\"of for every Lindel\"of $Y$ then $X$ is a $D$-space.

• A space $X$ has the Rothberger (Menger) property if for each sequence $\{U_n : n < \omega\}$ of open covers of $X$ (each closed under finite unions) for each $n$ there is a $U_n \in \cal{U}_n$ such that $\{U_n : n < \omega \}$ covers $X$.
• A space $X$ is $D$ if for each open neighborhood assignment $\{V_x : x \in X \}$ there is a closed discrete $D$ such that $\{C_x : x \in D\}$ covers $X$.

Tall PCF spaces

In this talk we examine the PCF space defined by Shelah and used by him to prove his celebrated theorem stating that if $\aleph_{\omega}$ is a strong limit cardinal then $2^{\aleph_{\omega} < \aleph_{\omega_4}$. This is achieved by showing that any PCF space must have height less than $\omega_4$. It is an open question whether this bound can be improved. We relate this problem to the study of thin tall locally compact scattered spaces and show that it is possible to have PCF spaces of height any ordinal less than $\omega_3$. We discuss some open problems and directions for further research.

Non-normality points

abstract in PDF format

Let $X$ be a discrete space, or more generally, a metrizable space. Let $\b X$ be the Stone-\v Cech comapctification of $X$, and let $y$ be a point in the remainder. Is $\b X \bs (x \cup \{y\})$ normal?

In the first section, I will review the bibliography of this question. Many of the contributors worked with Ken Kunen at Madison.

In the second section, I will consider the following situation: $X = \k \times I$, $I$ is the unit interval, $\k$ is a measurable cardinal with the discrete topology, $q$ is a $\k$-complete free ultrafilter on $\k$, and $y$ is the $q$-limit of the graph of the constant $0$ function.

In the third section, I will consider the following situation: $X = \k \times \mathbb{R}$, $\mathbb{R}$ is the real line, $\k$ is an infinite cardinal with the discrete topology, $p$ is a countably incomplete $\k^+$-good ultrafilter on $\k$, and $y$ is the $p$-limit of the graph of the constant $0$ function.

Tukey degrees of ultrafilters

This is joint work with Stevo Todorcevic. Given two ultrafilters $U$ and $V$ on the natural numbers, we say that $U$ is Tukey reducible to $V$ ($U\le_T V$) if there is a function $g:U\rightarrow V$ (called a Tukey map) which maps unbounded subsets of $U$ to unbounded subsets of $V$; equivalently, if there is a function $f:V \rightarrow U$ (called a cofinal map) which sends cofinal subsets of $V$ to cofinal subsets of $U$. Note that if $U$ is Rudin-Keisler reducible to $V$ then $U\le_T V$, but the converse is not in general true. We present some results on the structure of the Tukey degrees of ultrafilters, concentrating in particular on p-points, for which there is a special characterization of cofinal maps, and on ultrafilters with some properties in common with p-points.

How many sprays cover the plane?

A spray is a subset of the plane that has finite intersection withevery circle centered at a fixed point. I will present a generalresult that helps answering the question (due to J.H. Schmerl) in the title and also unifiesseveral other results in the literature concerning the relation betweenthe size of the continuum and the existence of certain coverings of theplane.

Countable dense homogeneity and Ungar's Theorems

All spaces are separable and metrizable. Ungar proved in 1978 that for a locally compact space $X$ such that no finite set separates it, the following statements are equivalent: (a) $X$ is Countable Dense Homogeneous (abbreviated CDH), $X$ is $n$-homogeneous for every $n$, and (c) $X$ is strongly $n$-homogeneous for every $n$. His main tool is the Effros Theorem on transitive actions of Polish groups on Polish spaces. We investigate whether this elegant result is optimal. The question whether one can prove a similar result with the assumption of local compactness relaxed to that of completeness is relevant in this context. We prove that for Polish spaces, a strong form of $n$-homogeneity for every $n$ is equivalent to a strong form of Countable Dense Homogeneity. In addition, we show that if $X$ is CDH and no set of size $n-1$ separates $X$, then $X$ is strongly $n$-homogeneous. No completeness assumptions are necessary here, the proof is elementary and does not need the Effros Theorem.

Continuous Metric AEC: superstability issues

I will describe some recent work on extensions of Continuous Model Theory to non-First Order contexts. I will describe several invariants crucial in superstability (or NIP), and some recent results of joint work with Pedro Zambrano.

Randomized Models and Continuous Logic

This lecture will present some results which will appear in a forthcoming joint paper by Itai Ben Yaacov and myself. The subject of continuous model theory was initiated by Chang and myself in the 1960's, and has recently been refined by Henson and others into a more useable form that looks much like current first order model theory. Given a first order structure M, a randomization of M is a continuous structure whose elements are random elements of M. Given a complete first order theory T, there is a corresponding complete theory T^R in continuous logic whose models are the randomizations of M. T^R has a natural set of axioms and admits elimination of quantifiers. Heuristically, the random elements of M behave much like the ordinary elements of M. This idea is captured by a series of results showing that many model theoretic properties hold for T^R if and only if they hold for T. These properties include separable categoricity, having a separable saturated model, having a prime model, omega-stability, and stability.

Some Applications of Set Theory to Geometry

We discuss some set-theoretic problems arising from concepts in geometric topology, such as local connectedness, and in plane geometry, such as smooth arcs. It is open whether PFA implies that every locally connected HL compactum is metrizable, but MA plus not CH does not. This is related to questions about covering uncountable subsets of Euclidean space by countably many smooth arcs.