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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 show that the product of a subparacompact C -scattered space and a Lindelof D-space is D. In addition, we show that every regular locally D-space which is the union of a finite collection of subparacompact spaces and metacompact spaces has the D-property. Also, we extend this result from the class of locally D-spaces to the wider class of D -scattered spaces. All the results are shown in a direct way.

Kiinzi, H.-P.A., and H.J.K. Junnila, Stability in quasi-uniform spaces and the inverse problem, Topology and its Applications 49 (1993) 175-189. We study the concept of a stable quasi-uniform space due to D. Doitchinov. In particular quasi-pseudo-metric spaces (X, d) whose associated quasi-uniformities %(d) or *U(d-‘) are stable are investigated.

In this paper the structure of hereditarily strong Σ-spaces (hsΣ-spaces, for short) is dealt with. The main result asserts that an hsΣ-space is the disjoint union of two σ subspaces one of which is an F σ , the other a G δ subset. Examples are given that in many ways, this decomposition cannot be improved. Then we investigate the question when an hsΣ-space is a σ-space. It is shown that a GO-space (or a first countable compactum) is metrizable iff it is an hsΣ-space, thereby proving a conjecture of J. van Wouwe. σ-spaces are characterized as being identical with perfect hsΣ-spaces. The question whether a Lindelof, first countable hsΣ-space is a σ-space is shown to be independent of set theory. A characterization of hsΣ-spaces with no compact subsets of cardinality >2 ω is given.

If the completely regular space X admits a quasi-perfect map onto the Dieudonne complete semi-metric space Y, then δX, the Dieudonne completion of X, is subparacompact; if in addition, Y is a Moore space, then δX is a subparacompact p-space.

The thrust of the paper is toward answering “What conditions on X and Y are sufficient to insure that X × Y has property Q?” Important results by section are 3. Suppose X is subcompact (has a σ-discrete closed cover of compact subsets). If X is collectionwise normal and Y is paracompact (metacompact, screenable), then X × Y is paracompact (metacompact, screenable). If X is T 2 and Y is subparacompact, then X × Y is subparacompact. 4. Let X be metrizable and Y have the G δ -property. If Y is screenable (metacompact, subparacompact), then X × Y is screenable (metacompact, subparacompact). 5. The product of a T 2 , paracompact, sequential space with a countably compact, normal space is collectionwise normal. 6. X × Y is an F-product if X is T 2 , paracompact, and sequential while Y is countably compact and normal, or if X is collectionwise normal and subcompact while Y is paracompact and normal.

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.

We will characterize metacompactness, subparacompactness and paracompactness of subspaces of products of two ordinal numbers. Using them we will show: 1. For such subspaces, weak submetaLindelofness, screenability and metacompactness are equivalent. 2. Metacompact subspaces of ω 1 2 are paracompact. 3. Metacompact subspaces of ω 2 2 are subparacompact. 4. There is a metacompact subspace of ( ω 1 +1) 2 which is not paracompact. 5. There is a metacompact subspace of ( ω 2 +1) 2 which is not subparacompact.