Applying Set Theory E88

We prove some results in set theory as applied to general topology and model theory. In particular, we study ℵ1-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets.

Normality Versus Paracompactness in Locally Compact Spaces

This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on ω1, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of ω1.

PFA(S)[S]: More Mutually Consistent Topological Consequences of PFA and V = L

Extending the work of Larson and Todorcevic, we show that there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable L-spaces or compact S-spaces. The model is one of the form PFA(S)[S], where S is a coherent Souslin tree.

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.