Yukinobu Yajima

A filter property of submetacompactness and its application to products

We show that there is a filter on ω such that for any sumetacompact space X and any open cover U of X, there is a sequence {Vn:nϵω} of open refinements of U such that {n: ord(x,Vn<gw} is in the filter for every x ϵ X. We apply this result to submetacompactness of product spaces, showing, e.g., that if X has a σ-closure-preserving cover by compact sets, then X × Y is submetacompact for every submetacompact space Y.

On ∑-products of semi-stratifiable spaces

Abstract Let ∑ be a ∑-product of semi-stratifiable spaces as in our title. First, we consider when ∑ is normal. Secondly, we consider when ∑ is collectionwise normal if it is normal. Moreover, it is shown that ∑ is subnormal.

Special refinements and their applications on products

Abstract We introduce a new concept, which is called special refinements. We illustrate that this concept plays certain roles to study normality of products. In fact, several known results concerning normality of products follow from our results as their corollaries.

Normality and countable paracompactness of products with σ-spaces having special nets

Abstract We introduce and study several subclasses of the class of σ-spaces. The smallest of the classes considered, that of LF-netted spaces, contains all F σ -discrete spaces and all stratifiable F σ -metrizable spaces. The main result of the paper establishes the equivalence of normality and countable paracompactness of the product of an LF-netted space with a countably paracompact and normal space.

Generalized paracompactness of subspaces in products of two ordinals

Abstract 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.

Analogous results to two classical characterizations of covering properties by products

Abstract First, as an analogue of Dowkers theorem for countable paracompactness, we prove a characterization of countable metacompactness in terms of subnormality of products. Second, as an analogue of Tamanos theorem for paracompactness, we give a characterization of Lindelofness in terms of normality of products.

Subnormality of X × κ and Σ-products

Abstract First, we prove that the product X × κ of a strong Σ-space X and a cardinal factor κ is (sub)shrinking if and only if it is (sub)normal. Secondly, we prove that a Σ-product of strong Σ-spaces is subshrinking if and only if it is subnormal. This answers a question raised by the author. Moreover, we discuss the subnormality of the product X × κ of a semi-stratifiable space X and a cardinal factor κ. Thus, we can find several similarities between X × κ and Σ-products.

Subparacompactness and submetacompactness of σ-products

Abstract In 1989, Chiba raised the problem of whether a σ-product of spaces, each finite subproduct of which is subparacompact (respectively, submetacompact), is subparacompact (respectively, submetacompact). In the present paper, we give an affirmative answer to this problem in each case.

Σ-products of paracompact C-scattered spaces

Abstract Many results have been obtained for the normality of Σ -products of generalized metric spaces. In this paper, we give another line for the study of the normality of Σ -products. That is, it is proved that a Σ -product of paracompact C -scattered spaces is (collectionwise) normal if it has countable tightness.

Characterizations of submetacompactness

Abstract 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.