Juris Steprāns

Continuous colourings of closed graphs

Abstract A topological version of a classical result of finite combinatorics concerning fixed point free functions is proved. This provides the motivation for considering the general question of when a closed graph on a topological space admits a colouring by a continous function.

MAXIMAL FILTERS, CONTINUITY AND CHOICE PRINCIPLES

Abstract In Zermelo-Fraenkel set theory ZF without the Axiom of Choice AC, results such as the following are established: The Boolean Prime Ideal Theorem BPI is equivalent to the statement: ZFE Every zero-filter is contained in a maximal one. The Boolean Prime Ideal Theorem is properly weaker than the statement: CFE Every closed filter is contained in a maximal one. The Axiom of Choice is equivalent to the conjunction of CFE and the Axiom of Countable Choice CC. The Axiom of Countable Choice is equivalent to the statement: C=SC Functions between metric spaces are continuous iff they are seqentially continuous.

Decomposing Euclidean space with a small number of smooth sets

Let the cardinal invariant fin denote the least number of continuously smooth n-dimensional surfaces into which (n + l)-dimensional Euclidean space can be decomposed. It will be shown to be consistent that fin is greater than Sn+l? These cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones.

Some results on CDH spaces—I

Abstract A space is CDH is any two countable dense sets can be mapped one onto the other by an autohomeomorphism of the entire space. The CDH nature of separable manifolds and R κ is examined.

MASAS IN THE CALKIN ALGEBRA WITHOUT THE CONTINUUM HYPOTHESIS

Abstract Methods for constructing masas in the Calkin algebra without assuming the Continuum Hypothesis are developed.

Extraspecial p-groups

Abstract It is shown that there is a class of cardinals λ for which there are 2λ extraspecial p-groups of size λ without abelian subgroups of size λ. This improves results of Ehrenfeucht and Faber and answers questions of Tomkinson.

The number of translates of a closed nowhere dense set required to cover a Polish group

Abstract For a Polish group G let cov G be the minimal number of translates of a fixed closed nowhere dense subset of G required to cover G . For many locally compact G this cardinal is known to be consistently larger than cov ( M ) which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups cov G = cov ( M ) . For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces.

Mutually complementary families of ₁ topologies, equivalence relations and partial orders

We examine the maximum sizes of mutually complementary families in the lattice of topologies, the lattice of T1 topologies, the semi-lattice of partial orders and the lattice of equivalence relations. We show that there is a family of K many mutually complementary partial orders (and thus To topologies) on K and, using this family, build another family of K many mutually T1 complementary topologies on K. We obtain K many mutually complementary equivalence relations on any infinite cardinal K and thus obtain the simplest proof of a 1971 theorem of Anderson. We show that the maximum size of a mutually T1 complementary family of topologies on a set of cardinality K may not be greater than K unless co < K < 2c . We show that it is consistent with and independent of the axioms of set theory that there be lt2 many mutually T1 -complementary topologies on t1 using the concept of a splitting sequence. We construct small maximal mutually complementary families of equivalence relations. 1. HISTORY AND INTRODUCTION In 1936, Birkhoff published On the combination of topologies in Fundamenta Mathematicae [7]. In this paper, he ordered the family of all topologies on a set by letting T1 < T2 if and only if T1 C T2. He noted that the family of all topologies on a set is a lattice. That is to say, for any two topologies T and a on a set, there is a topology T A a which is the greatest topology contained in both T and a (actually T A a = T n a) and there is a topology T V a which is the least topology which contains both T and a. This lattice has a greatest element, the discrete topology, and a smallest element, the indiscrete topology, whose open sets are just the null set and the whole set. In fact, the lattice of all topologies on a set is a complete lattice; that is to say there is a greatest topology contained in each element of a family of topologies and there is a least topology which contains each element of a family of topologies. A sublattice of this lattice which contains all the Hausdorff spaces is the lattice of T1 topologies. This is also a complete lattice whose smallest element is the Received by the editors June 10, 1992. 1991 Mathematics Subject Classification. Primary 54A10, 06B05; Secondary 03E50, 54A35, 06A12, 08A30. This work has been supported by the Natural Sciences and Engineering Research Council of Canada. @ 1995 American Mathematical Society

Maximal chains inωω and ultrapowers of the integers

SummaryVarious questions posed by P. Nyikos concerning ultrafilters on ω and chains in the partial order (ω, <*) are answered. The main tool is the oracle chain condition and variations of it.

Restricted compactness properties and their preservation under products

Abstract A topological space is said to have a restricted compactness property if every cover of it by sets chosen from a restricted class of open sets has a finite subcover. One such example is the class of spaces with the property that every cover by clopen sets has a finite subcover. The question of when such spaces as well as others with restricted compactness properties retain these properties under products is considered.