We present a construction from ♢* of a first countable, regular, countably metacompact space with a closed discrete subspace that is not a . 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 is a perfect subparacompact space and is a countable collection of subparacompact Čech-scattered spaces, then the product is subparacompact and (2) If is a countable collection of metacompact Čech-scattered spaces, then the product 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.
For a space X and a regular uncountable cardinal κ ≤ |X| we say that κ ∈ D(X) if for each with |T| = κ, there is an open neighborhood W of Δ(X) such that |T - W| = κ. If then we say that X has a small diagonal, and if every regular uncountable κ ≤ |X| belongs to D(X) then we say that X has an H-diagonal. In this paper we investigate the interplay between D(X) and topological properties of X in the category of generalized ordered spaces. We obtain cardinal invariant theorems and metrization theorems...
The notion of -normality was introduced and studied by Arhangel’skii, Ludwig in 2001. Recently, almost -normal spaces, which is a simultaneous generalization of -normal and almost normal spaces, were introduced by Das, Bhat and Tartir. We introduce a new generalization of normality, namely weak -normality, in terms of -closed sets, which turns out to be a simultaneous generalization of -normality and -normality. A space is said to be weakly -normal (w-normal if for every pair of disjoint...
We prove two theorems that characterize tightness in certain products of fans in terms of families of integer-valued functions. We also define several notions of forcing that allow us to manipulate the structure of the set of functions from some cardinal θ to ω, and hence, the tightness of these products. These results give new constructions of first countable <θ-cwH spaces that are not ≤θ-cwH.
Classical analytic spaces can be characterized as projections of Polish spaces. We prove analogous results for three classes of generalized analytic spaces that were introduced by Z. Frolík, D. Fremlin and R. Hansell. We use the technique of complete sequences of covers. We explain also some relations of analyticity to certain fragmentability properties of topological spaces endowed with an additional metric.
Several classes of hereditarily normal spaces are characterized in terms of extending upper semi-continuous compact-valued mappings. The case of controlled extensions is considered as well. Applications are obtained for real-valued semi-continuous functions.