Density topology and approximate continuity

Abstract

Abstract. We shall show that the space of all approximately continuous functions with the topology of pointwise convergence is not homeomorphic to its category analogue.2000 AMS Classification: 54C35, 54C30, 26A15.

The quasi topology associated with a countably subadditive set function

“…The ordinary density topology on R" is strictly finer than the strong density topology (except of course for n = 1 where d^ = d\ A rather deep result of Goffman, Neugebauer, and Nishiura [19] asserts that the ordinary density topology on R" is completely regular, but not normal, whereas the strong density topology is not even regular for n > 1. It is also known (contained implicitly in Ridder [20]) that every connected open subset of R" is connected in the ordinary density topology, hence also in the strong density topology.…”

“…2) G(x, y) should tend to 0 as one of the variables x or y goes ( 17 ) Proceeding as in the analogous case of the ordinary density topology (Goffman, Neugebauer and Nishiura [19]), we merely choose 2 disjoint countable (hence polar, hence finely closed) sets A, B C X each of which is dense in X in the euclidean topology. A finely continuous function /:X-->[0,1] with /(A) ={0}, /(B) = {1} would have to be discontinuous everywhere in X in the euclidian topology, which is impossible since/is of Baire class 1.…”

Characteristic Functions and Products of Bounded Derivatives

“…The terms " ¿/-closed", " ¿-interior" ( ri-int), etc., refer to the Denjoy topology (density topology) on R. (See, e.g., [2], [5].) We say that a function is approximately continuous if and only if it is continuous relative to the Denjoy topology.…”

On the lifting property and disintegration of measures

“…1) The idea of defining a topology re from a lower density was first pointed out to us by J. Oxtoby. It has been exploited by many authors in one context or another ( [17] ; [18] ; [22] ; [23] ; [43] ; [60] ; in the papers just quoted the Dunford-Pettis theorem is given in the context of locally convex spaces). We also want to remark here that the mapping U->gu exhibited in the formulation of (DP) is an isomorphism of the Banach space £(L l (Z, ju), F') onto L£'(Z, y).…”