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The symbol TD(X) denotes the hyperspace of all nonempty totally disconnected compact subsets of a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. For a mapping between Hausdorff spaces f : X → Y, define the induced mapping TD(f) : TD(X) → TD(Y) by TD(f)(A) = f(A) (the image of A under f). In the current paper, we study the relationships between the condition f belongs to a class of mappings between Hausdorff spaces 필 and the condition TD(f) belongs to 필. [ABSTRACT FROM AUTHOR]

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]

Some sufficient conditions are presented for the continuity of the mapping f : ⟨ X , τ X * ⟩ → ⟨ Y , τ Y * ⟩ , where τ X * and τ Y * are the topologies induced by the local function on 푋 and 푌, resp. under the assumption that the mapping from ⟨ X , τ X ⟩ to ⟨ Y , τ Y ⟩ is continuous. Further, we consider open and closed functions in the cases in which the open (or closed) mapping is preserved through the domain and codomain "idealization". Through several examples, we illustrate that the considered conditions cannot be weakened. [ABSTRACT FROM AUTHOR]