CHAPTER I - INTRODUCTION

Abstract

This chapter presents set theory and points set theory in the Euclidean plane (E2). The theory of point sets in Euclidean spaces gives the simplest example of general topology and historically, the investigation of the former theory by G. Cantor in the late 19th century led to the establishment of the concept of topological space by F. Hausdorff, M. Frechet, C. Kuratowski, and the other mathematicians in the early 20th century. The concept of convergence of a point sequence is significant in E2, especially with respect to analysis in E2. This concept is closely related with other concepts such as neighborhood, closure, and open set. An empty or vacant set is a set that contains no elements and is denoted by Ø. The chapter discusses important properties of open sets: E2 and Ø are open sets, the intersection of finitely many open sets is open, and the union of open sets is open. The properties of closed sets include E2 and Ø are closed sets, the union of finitely many closed sets is closed, and the intersection of closed sets is closed.