Characterizations of K- Semimetric Spaces

In this paper, we prove, for a space X, the following are equivalent:1. X is a D1 space with a regular-Gδ-diagonal,2. X is a D2 space with a regular-Gδ-diagonal, 3. X is a semi-developable space with Gδ (3) -diagonal, 4. X is a D1-space with a Gδ(3)-diagonal, 5. X is a D2 -space with a Gδ(3)-diagonal, 6. X is a q, -space with a G*δ (2)-diagonal, 7. X is a semi-developable space with G*δ (2)-diagonal, 8. X is a semimetrizable, c-stratifiable space, 9. X is a c-Nagata -space, 10. X is a K-semimetrizable.

The Normality in Products with a Countably Compact Factor

It is known that the product ω1 × X of ω1 with an M1-spacemay be nonnormal. In this paper we prove that the product κ × X of an uncountable regular cardinal κ with a paracompact semi-stratifiable space is normal iff it is countably paracompact. We also give a sufficient condition under which the product of a normal space with a paracompact space is normal, from which many theorems involving such a product with a countably compact factor can be derived.