Advanced Mathematics

5.4Point‐Set Topology on the Real Line

Purpose of Section

To introduce some topological concepts of the real number system, such as open and closed sets, interior, boundary, and exterior points of a set, and limit points. These act as the foundation of many concepts in analysis.

5.4.1 Introduction

There are several areas of topology, such as graph theory and network topology, geometric topology, algebraic topology, differential topology, combinatorial topology, and point‐set (or general) topology. Some areas of topology are sufficiently diverse that practicing researchers in one area may have only a passing knowledge of other areas. While graph theory is interested in connections between objects, and geometric topology was interested in shapes of objects, point‐set topology is about “closeness.” By closeness, we mean closeness of numbers, points, functions, matrices, operators, and other mathematical objects. The interest in closeness lies in the fact it gives rise to limits, infinite series, convergence, continuity, and mathematical concepts associated with calculus, differential equations, real and complex analysis, as well as areas of science and engineering.

Point‐set topology can be studied in any dimension, even infinite, but in this section, we restrict our attention to the one‐dimensional real line, mostly because it is easier to visualize than in two and three dimensions, and a whole lot simpler than dimensions larger than three.¹ The basic concepts of point‐set ...