We present a construction from ♢* of a first countable, regular, countably metacompact space with a closed discrete subspace that is not a $G_{\delta }$. In addition some nonperfect spaces with σ-disjoint bases are constructed.

In this paper, we discuss covering properties in countable products of Čech-scattered spaces and prove the following: (1) If $Y$ is a perfect subparacompact space and $\{X_n:n\in \omega \}$ is a countable collection of subparacompact Čech-scattered spaces, then the product $Y\times {\prod }_{n\in \omega }{X}_n$ is subparacompact and (2) If $\{X_n:n\in \omega \}$ is a countable collection of metacompact Čech-scattered spaces, then the product ${\prod }_{n\in \omega }{X}_n$ is metacompact.

2000 Mathematics Subject Classification: 54C10, 54D15, 54G12.For given completely regular topological spaces X and Y, there is a completely regular space X ~⊗ Y such that for any completely regular space Z a mapping f : X × Y ⊗ Z is separately continuous if and only if f : X ~⊗ Y→ Z is continuous. We prove a necessary condition of normality, a sufficient condition of collectionwise normality, and a criterion of normality of the products X ~⊗ Y in the case when at least one factor is scattered.