Ronnie Levy

Cofinality in normal almost compact spaces

This is the published version, also available here: http://dx.doi.org/10.1090/S0002-9939-1991-1072087-1#sthash.G2e2uNs1.dpuf. First published in Proc. AMS. in 1991, published by the American Mathematical Society.

Separation in Ψ-spaces

Abstract A Ψ -space is the topological space usually associated with a maximal almost disjoint family of subsets of the integers. In this paper, we study properties that an infinite set of nonisolated points in a Ψ -space might possess. These properties are 2-embeddedness, C ∗ -embeddedness, and another weaker property, that of being solidly normalized. It is known from [2,3] that the existence of a Ψ -space with an infinite set of nonisolated points with any of these properties is independent of ZFC. Here we show that these three properties are distinct by providing examples, assuming less than Martins axiom, of Ψ -spaces with infinite sets of nonisolated points which in one case are C ∗ - but not 2-embedded, and in the other case solidly normalized but not C ∗ -embedded. Additionally, Martins axiom implies the existence of a Ψ -space with a set S of nonisolated points of cardinality c such that every subset of S with cardinality less than c is 2-embedded.

Ordered spaces all of whose continuous images are normal

This is the published version, also available here: http://dx.doi.org/10.1090/S0002-9939-1989-0973846-4. First published in Proc. AMS. in 1989, published by the American Mathematical Society.

Techniques and examples in U-embedding

Abstract A subset S of a metric space X is U -embedded in X if every uniformly continuous real-valued function on S extends to a uniformly continuous real-valued function on X . In this paper, techniques are presented which allow us to determine whether certain subsets of various metric spaces are U -embedded. Examples are given which indicate the difficulty of showing which sets are U -embedded.

Not realcompact images of not Lindelöf spaces

Abstract X has not a realcompact image if X is not Lindelof and at least one of the following: X has Lindelof degree at most 2 ω , X is countability tight, X maps continuously onto X 2 . If X has an uncountable closed discrete subset, then X has a one-to-one not realcompact continuous image. If X is not Lindelof, then X , the sum of the finite powers of X , has a not realcompact continuous image; hence L ( X ) = pq ( pq ) = t ( C p ( X )).

Pseudocompactness and extension of functions in Franklin-Rajagopalan spaces

Abstract Compactifications of the set of integers whose remainders are [0, ω 1 ] are studied. Conditions assuring the existence of such spaces with certain pseudocompact subspaces are given. Extension of functions on N in such compactifications is discussed.

Some problems from George Mason University

Publisher Summary This chapter discusses some of Kuleszas problems, Levys problems, and Matveevs problems raised at George Mason university. The behavior of dimension for nonseparable metric spaces is not well understood, despite having been studied for well over half a century; there are no analogs for several important theorems regarding dimension in separable spaces and generally the results and examples are quite complicated. Almost any new theorem or example relating to the covering dimension dim would be interesting; there are several problems in the chapter that are of interest. The focus is on two fundamental problems that remain largely unsolved. The relatively recent remarkable example νμ0 of Mrowka gives a consistent solution to one of the great problems in dimension theory. Its finite powers give examples of metric spaces for which dim-ind, the discrepancy between covering and the small inductive dimension, can be any positive integer. The chapter elaborates problems in dimension theory of nonseparable metric spaces. A question about weak P-points is also discussed in the chapter.

Remainders of normal spaces

Abstract Fleissner, W., J. Kulesza and R. Levy, Remainders of normal spaces, Topology and its Applications 49 (1993) 167-174. If X is totally compact, then X is the Stone-tech remainder of a normal space. A partial converse: if X is first countable and the Stone-tech remainder of a normal space, then X is locally compact. Every metric space, but not every first countable space, is the remainder of a normal space. For countable spaces, or even countable spaces which are locally compact except at one point, there are examples, but few theorems. We show that a construction of Porter and Woods applies to certain examples only if b = b. We investigate the property that all normal images are compact, and show that large products minus small subsets have this property. Keywords: Remainder, normal, locally compact, countable type, Stone-tech, Z-product, count- able spaces.

Compactifications of the rationals and small compact spaces

Abstract Conditions assuring that a compact space is a compactification of the rationals are given. Relations between the π-weight and strong density are discussed.