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We show that if X is a regular D ¯ -scattered space and X is the union of a finite collection of submetacompact spaces, then X is a D-space, where D ¯ is the class of all D-spaces, hence X is a D-space if X is the union of a finite collection of submetacompact C -scattered spaces, where C is the class of all compact spaces. This gives a positive answer to a question mentioned by Martinez [J.C. Martinez, 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 θ ¯ -refinable (or (sub)metacompact) C -scattered spaces and a Lindelof D-space is a D-space. In the last part of this note, we show that if X is a regular meta-Lindelof D ¯ -scattered space with locally countable extent, then X is a D-space. As a corollary, every meta-Lindelof locally compact Hausdorff space is a D-space.

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, ω 1 -compactness = Lindelofness; 2) on weakly subparacompact Hausdorff spaces with rudimentary normality, regularity = normality = countable paracompactness; and 3) on weakly subparacompact regular T 1 -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.

We give an internal characterization of submetacompactness and then we use it to prove that a space X is submetacompact if and only if X × ( κ + 1) is suborthocompact, where κ is a cardinal no less than the Lindelof degree of X . Similarly, we also obtain that a space X is metacompact if and only if X × ( κ + 1) is orthocompact.

In this paper, we shall discuss submetacompactness and weak submetacompactness in countable products of Čech-scattered spaces and prove the following: (1) If $\{ X_{n} : n \in \omega \}$ is a countable collection of submetacompact Čech-scattered spaces, then the product $\Prod_{n\in \omega} X_{n}$ is submetacompact. (2) If $Y$ is a hereditarily weakly submetacompact space and $\{X_{n} : n \in \omega \}$ is a countable collection of weakly submetacompact Čech-scattered spaces, then the product $Y \times \Prod_{n\in \omega}X_{n}$ is weakly submetacompact.

We prove that a locally connected, rim-Lindelof, submeta Lindelof, normal, ⩽ω 2 -collectionwise T 2 space is paracompact. Further, if ‘Lindelof’ is replaced by ‘compact’, then ‘⩽ω 2 -collectionwise T 2 ’ can be omitted.

It is proven that under V = L, every locally compact, countably paracompact space is collectionwise normal with respect to compact sets. From this it can be shown that under V = L, submetacompact implies paracompact in the class of locally compact, countably paracompact spaces.

In the present paper, the four covering properties of topological spaces are paracompactness, subparacompactness, metacompactness and submetacompactness. We characterize these with the properties of certain products in terms of open (or closed) rectangles.