Chapter 15 - Density Topologies

Abstract

This chapter presents basic properties of the density topology on Euclidean spaces with special emphasis on the one-dimensional case. The properties of single density points of a set and the lower density operator are provided in the chapter. The density theorem is often proved by means, which are more or less equivalent to Vitali's covering theorem. It is observed that because a density topology is strictly greater than a natural topology, the classes of connected sets fulfill the converse inclusion and the equality follows. There are other topologies related to the density topology and some subclasses of the density family of approximately continuous functions. The class of all measurable functions coincides with the union of all the classes of functions, which are approximately continuous almost everywhere. The family of all density topologies is partially ordered by inclusion. The structure of this partially ordered set is rather complicated. The local properties of measurable sets are also presented in the chapter.