s-point finite refinable spaces

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.

Paracompactness in Locally Lindelöf Spaces

This paper contains a set of results concerning paracompactness of locally nice spaces which can be proved by (variations on) the technique of “stationary sets and chaining” combined with other techniques available at the present stage of knowledge in the field. The material covered by the paper is arranged in three sections, each containing, in essence, one main result.The main result of Section 1 says that a locally Lindelöf, submeta-Lindelöf ( = δθ-refinable) space is paracompact if and only if it is strongly collectionwise Hausdorff. Two consequences of this theorem, respectively, answer a question raised by Tall [7], and strengthen a result of Watson [9]. In the last two sections, connected spaces are dealt with. The main result of the second section can be best understood from one of its consequences which says that under 2ωl > 2ω, connected, locally Lindelöf, normal Moore spaces are metrizable.

Supports of Borel measures

We present a new class of topological spaces called SL-spaces, on which every Borel measure has a Lindelöf support. The class contains all metacompact spaces. However, a θ-refinable space is not necessarily an SL-space.

On irreducible spaces

A constructive proof of a theorem of Worrell and Wicke, which states that every open covering of a θ-refinable space has a σ-distributively point-finite open refinement that covers the space minimally, is presented. Spaces for which every open covering has an open refinement which covers the space minimally are characterized by the use of discrete closed collections and some open questions relating to spaces of this type are included.