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]

Let K and S be locally compact Hausdorff spaces and let T be a real linear isomorphism of an extremely regular subspace A of C_{0}(K) into C_{0}(S) satisfying \|T \| \ \|T^{-1}\|<2. Put L=T \ \|T^{-1}\| and pick 0 \leq \varepsilon <1 such that \|T \| \ \|T^{-1}\|=1+\varepsilon. We prove that there exist a subset S_0 of S, a proper map \varphi of S_{0} onto K and a map \lambda :S_{0} \to \{-1, 1\} such that \begin{equation*} |Lf(s)-\lambda (s) f(\varphi (s))| \leq \varepsilon \|f\|, \ \forall s \in S_{0} \ \text {and} \ f \in A. \end{equation*} For \varepsilon < 1/2 the maps \varphi and \lambda are continuous, and this approximation of L by such a weighted composition operator improves the well-known Holsztyński theorem [Studia Math. 26 (1966), pp. 133–136], the case where \varepsilon =0, K and S are compact and A=C_{0}(K). The approximation of L in the general case where 0 \leq \varepsilon <1 is an extension of a Benyamini's result [Proc. Amer. Math. Soc. 83 (1981), pp. 479–485] which allowed us to strengthen a classical Jarosz's theorem [Proc. Amer. Math. Soc. 90 (1984), pp. 373–377] by proving that K is a continuous image not just of a subset of S but of a locally compact subset of S. Moreover, when K is compact, it is a continuous image not just a closed subset of S but of a compact subset of S, namely the set S_0 having the above properties. [ABSTRACT FROM AUTHOR]