Search

A study of the class \Delta consisting of topological \Delta-spaces was originated by Jerzy Ka̧kol 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 \Delta _2 \subset \Delta _1 properly containing \Delta. We observe that for every first-countable X the following equivalences hold: X\in \Delta _1 iff X\in \Delta _2 iff each countable subset of X is G_{\delta }. Thus, new proposed concepts provide a natural extension of the family of all \lambda-sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X belongs to the class \Delta _1 iff countable subsets of X are scattered. (2) Every regular scattered space belongs to the class \Delta _2. We investigate whether the classes \Delta _1 and \Delta _2 are invariant under the basic topological operations. Similarly to \Delta, both classes \Delta _1 and \Delta _2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to \Delta, they are not preserved by closed continuous images. Let Y be l-dominated by X, i.e. C_p(X) admits a continuous linear map onto C_p(Y). We show that Y \in \Delta _1 whenever X \in \Delta _1. 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 \beta Y is scattered. [ABSTRACT FROM AUTHOR]

A remainder of the Hilbert space l2 is a space homeomorphic to Z\l2, where Z is a metrizable compact extension of l2, with l2 dense in Z. We prove that for any remainder K of l2, every non-one-point closed image of K contains either a compact set with no transfinite dimension or compact sets of arbitrarily high inductive transfinite dimension ind. We shall also construct for each natural n a σ-compact metrizable n-dimensional space whose image under any non-constant closed map has dimension at least n and analogous examples for the transfinite inductive dimension ind (this provides a rather strong negative solution of a problem in [Dissertationes Math. (Rozprawy Mat.) 216 (1983), pp. 1-41]). [ABSTRACT FROM AUTHOR]