SOME CLASSES OF TOPOLOGICAL SPACES EXTENDING THE CLASS OF Δl-SPACES

A study of the class Δl consisting of topological Δl-spaces was originated by Jerzy K,akol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86-99; Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267-280]. The main purpose of this paper is to introduce and investigate new classes Δl2 ⊂ Δl1 properly containing Δl. We observe that for every first-countable X the following equivalences hold: X ∈ Δl1 iff X ∈ Δl2 iff each countable subset of X is Gδ. Thus, new proposed concepts provide a natural extension of the family of all λ-sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X belongs to the class Δl1 iff countable subsets of X are scattered. (2) Every regular scattered space belongs to the class Δl2. We investigate whether the classes Δl1 and Δl2 are invariant under the basic topological operations. Similarly to Δl, both classes Δl1 and Δl2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to Δl, they are not preserved by closed continuous images. Let Y be l-dominated by X, i.e. Cp(X) admits a continuous linear map onto Cp(Y ). We show that Y ∈ Δl1 whenever X ∈ Δl1. Moreover, we establish that if Y is l-dominated by a compact scattered space X, then Y is a pseudocompact space such that its Stone-Čech compactification βY is scattered.