CHAPTER XI - MEASURE IN THE PLANE AND IN SPACE

Abstract

This chapter discusses the measure of plane and space sets. An open interval in the plane will be any open rectangle with sides parallel to the coordinate axes, and a closed interval will be the same rectangle with its boundary. Understanding a plane The chapter discusses an open set, limit point, and a closed set in the plane formally in the same way this was done on the real line. The complement of an open set is a closed set and conversely; the union of an arbitrary number and the intersection of a finite number of open sets are open sets; the intersection of an arbitrary number and the union of a finite number of closed sets are closed sets. The variety of open sets in the plane is greater than on the real line. It is not true that every open set is the union of a finite or denumerable number of disjoint open intervals, as is the case of the real line.