The lattice generated by τ-quasicontinuous functions

Abstract

The lattices generated by the families of quasicontinuous and τ -quasicontinuous functions are described. Keywords Functions quasicontinuous · Discrete limitsWe point out that, except for the case where a topology T is specifically mentioned, all topological notions are given with respect to the natural topology.Throughout the paper, R and N will denote the sets of all real numbers and of all positive integers, respectively, I the σ -ideal of subsets of R of the first category, S the σ -field of subsets of R having the Baire property. For any a ∈ R and A ⊂ R we denote a · A = {ax : x ∈ A} and A − a = {x − a : x ∈ A}. Recall [7] that 0 is an I-density point of set A ∈ S if and only if, for every increasing sequence {n m } m∈N of positive integers, there exists a subsequence {n m p } p∈N such that χ n m p ·A∩[−1,1] (x) −→ 1 as p −→ ∞ Communicated by P. de Lucia.