Paul J. Szeptycki

On some classes of Lindelöf Σ-spaces

Abstract We consider special subclasses of the class of Lindelof Σ-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space X is in the class L Σ ( ⩽ κ ) if it admits a cover by compact subspaces of weight κ and a countable network for the cover. We restrict our attention to κ ⩽ ω . In the case κ = ω , the class includes the class of metrizably fibered spaces considered by Tkachuk, and the P-approximable spaces considered by Tkacenko. The case κ = 1 corresponds to the spaces of countable network weight, but even the case κ = 2 gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight ℵ 1 is in the class L Σ ( ⩽ ω ) , answering a question of Tkachuk. As well, we study whether certain compact spaces in L Σ ( ⩽ ω ) have dense metrizable subspaces, partially answering a question of Tkacenko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.

κ-normality and products of ordinals

A regular topological space is called κ-normal if any two disjoint regular closed subsets can be separated. In this paper we will show that any product of ordinals is κ-normal. In addition a generalization of a theorem of van Douwen and Vaughan will be proven and used to give an alternate proof that the product of any countable family of ordinals is κ-normal.  2001 Elsevier Science B.V. All rights reserved. MSC: 54B10; 54D15; 54D20; 03E10; 03E75

Soft almost disjoint families

An almost disjoint family A is said to be soft if there is an infinite set that meets each a ∈ A in a nonempty but finite set. We consider the associated cardinal invariant defined to be the minimal cardinality of an almost disjoint family that is not soft. We show that this cardinal coincides with J. Brendles cardinal ap.

Transversals for strongly almost disjoint families

For a family of sets A, and a set X, X is said to be a transversal of A if X C ∪ A and |a ∩ X | = 1 for each a ∈ A. X is said to be a Bernstein set for A if ∅ ≠ a ∩ X ≠ a for each a ∈ A. Erdos and Hajnal first studied when an almost disjoint family admits a set such as a transversal or Bernstein set. In this note we introduce the following notion: a family of sets A is said to admit a σ-transversal if A can be written as A = ∪{A n : n ∈ ω} such that each An admits a transversal. We study the question of when an almost disjoint family admits a σ-transversal and related questions.

\({G_\delta}\) covers of compact spaces

We solve a long standing question due to Arhangel’skii by constructing a compact space which has a

Spaces of continuous functions defined on Mrówka spaces

{G_\delta}

Fréchet-Urysohn for finite sets, II

Gδ cover with no continuum-sized (