Chapter 3
Topology of the Real
Number System
Sections 3. 1 and 3.2 present the concepts of neighborhoods,
open and closed sets, interior and boundary points, cluster
points, and closures, which are essential tools in modern real
analysis. Sections 3.3 and 3.4 can be safely omitted in a one-
semester course. For the core of this book, a compact set is
one that is closed and bounded, although the open covering
approach is given full treatment in Section 3.3.Mathematical topics often have both algebraic and geometric (or visual) sides.
Indeed, students and instructors often find visualization to be an indispensable
tool in learning and remembering new mathematical concepts. In this chapter
we introduce a powerful geometric tool, called "topology," that has proved
invaluable in formulating the ideas of elementary real analysis. It introduces
a language that is highly suggestive of visualization. By its very nature, it is
qualitative rather than quantitative. Topology is a subject in its own right.
Here we barely scratch the surface of this wide and deep subject. It is difficult
t o say in a few words just what "topology" is and what its achievements have
been. To give a complete definition would require us to digress too far from our
objective. Briefly, it is a geometric type of mathematics in which neither size nor
shape has any significance. In fact, one can say that "distance,'' so important in
the concepts of ordinary geometry, plays no essential role in topology. We shall
shortly define what is meant by an "open set" in the real number system. Once
we know what an open set is, distance will no longer be necessary in defining
limits, continuous functions, and other elementary concepts in analysis. The
concept of "open" is as basic to topology as "distance" is to ordinary geometry.
Some knowledge of elementary topology is indispensable in learning modern
137