CHAPTER ONE - TOPOLOGICAL SPACES

Abstract

This chapter focuses on topological spaces. A topological space can be defined by taking closed set as a primitive term instead of closure and supposing that the following axioms are satisfied: (1) the union of a finite family of closed sets and (2) the intersection of an arbitrary family of closed sets is a closed set. In the space of integers, every set is simultaneously closed and open. The intersection of a finite family of open sets and the union of an arbitrary family of open sets is an open set. Topological spaces with a countable local base at each point are of great importance. In the space of natural numbers, the boundary of every set is empty. In the space of real numbers, the boundary of the set of rational numbers is the entire space.