Masami Sakai

The Combinatorics of Open Covers

The combinatorics of open covers is a study of Cantor’s diagonal argument in various contexts. The field has its roots in a few basic selection principles that arose from the study of problems in analysis, dimension theory, topology and set theory. The reader will also find that some familiar works are appearing in new clothes in our survey. This is particularly the case in connection with such problems as determining the structure of compact scattered spaces and a number of classical problems in topology. We hope that the new perspective in which some of these classical enterprises are presented will lead to further progress. In this article we also attempt to give the reader an overview of the problems and techniques that are currently fueling much of the rapidly increasing current activity in the combinatorics of open covers.

Topological groups with a certain point-countable cover

Abstract Shibakov proved in 1998 that every sequential topological group with a point-countable k -network is metrizable if its sequential order is less than ω 1 . In this paper, we study further topological properties of sequential topological groups with a point-countable k -network. We introduce the notion of a cs -cover, and show that: If G is a sequential topological group with a point-countable cs -cover F such that any product of finitely many elements of F is Lindelof, then G has an open subgroup which is the union of countably many elements of F . In addition, we show that every topological group with a σ -hereditarily closure-preserving k -network is an ℵ -space.

Variations on tightness in function spaces

Abstract For a Tychonoff space X we denote by C p ( X ) the space of all real-valued continuous functions on X with the topology of pointwise convergence. It is known that some topological properties of C p ( X ) can be well characterized by topological properties of X. For example, Arhangelskii and Pytkeev proved that the tightness of C p ( X ) is countable if and only if every finite power of X is Lindelof. In this paper, we give topological properties of X which characterize T-tightness and set-tightness of C p ( X ). A characterization of set-tightness of C p ( X ) answers a question posed by Bella.

Non-(ω, ω1)-regular ultrafilters and perfect κ-normality of product spaces

Abstract In this paper we give a topological characterization of non-(ω, ω 1 )-regular ultrafilters by means of perfect κ-normality of product spaces. For a countably incomplete uniform ultrafilter p on λ > ω, we prove that p is not (ω, ω 1 )-regular iff Y p × D τ is perfectly κ-normal for each cardinal τ, where D = {0, 1} and Y p is the subspace consisting of the point p and all isolated points in β D (λ). As corollaries, (1) Y p × D ω 1 is not perfectly κ-normal for each uniform ultrafilter p on ω 1 , if we assume V = L or MA ω 1 , (2) Woodins model contains a uniform ultrafilter p on ω 1 such that Y p × D τ is perfectly κ-normal for each cardinal τ.

Menger subsets of the Sorgenfrey line

A space X is said to have the Menger property if for every sequence {u n : n ∈ ω} of open covers of X, there are finite subfamilies V n ⊂ U n (n, ∈ ω) such that U n∈ω V n is a cover of X. Let i: S → R be the identity map from the Sorgenfrey line onto the real line and let X S = i -1 (X) for X ⊂ ℝ. Lelek noted in 1964 that for every Lusin set L in ℝ, L S has the Menger property. In this paper we further investigate Menger subsets of the Sorgenfrey line. Among other things, we show: (1) If X S has the Menger property, then X has Marczewskis property (s 0 ). (2) Let X be a zero-dimensional separable metric space. If X has a countable subset Q satisfying that X \ A has the Menger property for every countable set A C X \ Q, then there is an embedding e: X → R such that e(X) S has the Menger property. (3) For a Lindelof subspace of a real GO-space (for instance the Sorgenfrey line), total paracompactness, total metacompactness and the Menger property are equivalent.

Perfect κ-Normality of Product Spaces

ABSTRACT. A space X is called perfectly K‐normal (respectively, Klebanov) if the closure of every open set (respectively, every union of zero‐sets) in X is a zero‐set. It is proved: The product of infinitely many Lašnev spaces need not be perfectly K‐normal, in particular, S(ω1)2χ Dω1 is not perfectly K‐normal; a locally compact, paracompact space Y is Klebanov if and only if XχY is perfectly K‐normal for every Lašnev space X; if XχY is perfectly K‐normal for every paracompact s̀‐space X, then Y is perfectly normal. Properties of a Klebanov space are also studied.

Notes on strongly Whyburn spaces

We introduce the notion of a strongly Whyburn space, and show that a space

Some weak covering properties and infinite games

X

Subparacompactness and submetacompactness of σ-products

is strongly Whyburn if and only if

Embeddings of κ-metrizable spaces into function spaces

X\times(\omega+1)