The Intersection Conditions For 〈<em>s</em>〉-Density Systems of Paths

Abstract

AbstractWe investigate the intersection conditions for an hsi-density systemof paths. We show, for example, that for every unbounded and nonde-creasing sequence of positive numbers hsi such that liminf n!1s n s n+1 =0, there exists a system of paths connected with hsi-density points whichdoes not satisfy the intersection conditions. Moreover, we show that afunction f : R ! R is hsi-approximately continuous if and only if f iscontinuous with respect to some hsi-density system of paths. 1 Introduction. The notion of a density point was introduced by Lebesgue at the beginningof the 20th century. Together with his fundamental theorem that almost allpoints of a Lebesgue measurable set are density points of that set, they turnedout to play an important role in the theory of real functions.The density topology connected with the notion of a density point wasdiscovered by Haupt and Pauc [5] and was subsequently studied in detailby Go man, Neugebauer, Nishiura, Waterman and Tall. It turned out thata function is approximately continuous if and only if it is continuous withrespect to the density topology.Some generalizations of the notion of a Lebesgue density point on the realline were introduced by Taylor [8]. Wilczynski presented the concept of adensity point with respect to category [10]. Further, Filipczak and Hejduk [3],generalized the notion of a density point using unbounded and nondecreasingsequences of positive numbers. Such an approach to density points gave riseto hsi-density topologies and hsi-approximately continuous functions. Some