Some Modifications of the Core Topology on the Plane

Abstract

In this paper we introduce some modifications of the core topology on the plane. Here the role of the Euclidean topology on the real line on every or almost every direction is played by the density topology or the Hashimoto topology connected with the $\sigma$-ideal of null sets and the $\sigma$-ideal of meager sets. We demonstrate the proper inclusions between these families, we investigate the separation axioms and functions continuous with respect to these topologies.