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We first point out that the product of a subparacompact D C -like space and a subparacompact D -space is a D -space. As a main result, we use elementary submodels to prove that any countable product of subparacompact D C -like spaces is a D -space. More generally, we prove that any uncountable product of subparacompact D C -like spaces is a D -space if and only if all but countably many of the factors are compact. [ABSTRACT FROM AUTHOR]

Abstract: We show that if X is a regular -scattered space and X is the union of a finite collection of submetacompact spaces, then X is a D-space, where is the class of all D-spaces, hence X is a D-space if X is the union of a finite collection of submetacompact -scattered spaces, where is the class of all compact spaces. This gives a positive answer to a question mentioned by Martínez [J.C. Martínez, On finite unions and finite products with the D-property, Topology Appl. 158 (2) (2011) 223–228]. In the second part of this note, we show that the product of a finite collection of regular weak -refinable (or (sub)metacompact) -scattered spaces and a Lindelöf D-space is a D-space. In the last part of this note, we show that if X is a regular meta-Lindelöf -scattered space with locally countable extent, then X is a D-space. As a corollary, every meta-Lindelöf locally compact Hausdorff space is a D-space. [Copyright &y& Elsevier]

Abstract: We identify some remnants of normality and call them rudimentary normality, generalize the concept of submetacompact spaces to that of a weakly subparacompact space and that of a weakly⁎ subparacompact space, and make a simultaneous generalization of collectionwise normality and screenability with the introduction of what is to be called collectionwise σ-normality. With these weak properties, we show that, 1) on weakly subparacompact spaces, countable compactness=compactness, -compactness=Lindelöfness; 2) on weakly subparacompact Hausdorff spaces with rudimentary normality, regularity=normality=countable paracompactness; and 3) on weakly subparacompact regular -spaces with rudimentary normality, collectionwise σ-normality=screenability = collectionwise normality=paracompactness. The famous Normal Moore Space Conjecture is thus given an even more striking appearance and Worrell and Wickeʼs factorization of paracompactness (over Hausdorff spaces) along with Krajewskiʼs are combined and strengthened. The methodology extends itself to the factorization of paracompactness on locally compact, locally connected spaces in the manner of Gruenhage and on locally compact spaces in that of Tall, and to the factorization of subparacompactness and metacompactness in the genre of Katuta, Chaber, Junnila and Price and Smith and that of Boone, improving all of them. [Copyright &y& Elsevier]