S-point finite refinable spaces

Sheldon W Davis,Elise M Grabner,Gray C Grabner

doi:10.1155/s0161171299223678

Abstract

A spaceXis calleds-point finite refinable (ds-point finite refinable) provided every open cover𝒰ofXhas an open refinement𝒱such that, for some (closed discrete)C⫅X,(i) for all nonemptyV∈𝒱,V∩C≠∅and(ii) for alla∈Cthe set(𝒱)a={V∈𝒱:a∈V}is finite.In this paper we distinguish these spaces, study their basic properties and raise several interesting questions. Ifλis an ordinal withcf(λ)=λ>ωandSis a stationary subset ofλthenSis nots-point finite refinable. Countably compactds-point finite refinable spaces are compact. A spaceXis irreducible of orderωif and only if it isds-point finite refinable. IfXis a strongly collectionwise Hausdorffds-point finite refinable space without isolated points thenXis irreducible.

Highlights

(i) for all nonempty V ∈ ᐂ, V ∩ C ≠ ∅ and (ii) for all a ∈ C the set (ᐂ)a = {V ∈ ᐂ : a ∈ V } is finite
We show that every space X can be embedded as a closed subset of an irreducible space, Theorem 1.3
We do not know if every ds-point finite refinable space is irreducible

Summary

Introduction

It follows from [6, Thm. 1.1], that a space X is irreducible if and only if every open cover ᐁ has a C-point finite open refinement ᐂ for some closed discrete set C ⊆ X such that for all a ∈ C, |ᐂa| = 1. A space is metacompact if and only if every monotone open cover has a point finite open refinement, [17].

Results

Conclusion