Intuitive Concepts in Elementary Topology

B.H. Arnold

• Publisher: Dover Publications
• Year: 2015
• ISBN:
  9780486275765
• Topic: Mathematik
• Subtopic: Topologie

Classroom-tested and much-cited, this concise text offers a valuable and instructive introduction for undergraduates to the basic concepts of topology. It takes an intuitive rather than an axiomatic viewpoint, and can serve as a supplement as well as a primary text.
A few selected topics allow students to acquire a feeling for the types of results and the methods of proof in mathematics, including mathematical induction. Subsequent problems deal with networks and maps, provide practice in recognizing topological equivalence of figures, examine a proof of the Jordan curve theorem for the special case of a polygon, and introduce set theory. The concluding chapters examine transformations, connectedness, compactness, and completeness. The text is well illustrated with figures and diagrams.

EIGHT

Spaces

8-1 Introduction

In Chapter 7 we have seen how the concept of the distance between two points in three-dimensional space can be used to define continuity of transformations; this concept, in turn, was used to define topological equivalence of figures. But we have also discussed some figures (for example, the Klein bottle) which are not subsets of three-dimensional Euclidean space. The definitions of continuity and topological equivalence can be generalized so that they are applicable in these situations. The generalization is made in this chapter.

In Section 8-2 we show that the concept of the distance between two points may be available even when the “points” are elements of an arbitrary set (perhaps a set of functions). In this case, we speak of the set as a metric space; the definitions of continuity and topological equivalence can be carried over to metric spaces immediately from their statements in Chapter 7.

In Section 8-3 we shall find that some of the concepts which can be defined in a metric space (e.g., open set, closed set) may be available in still more general situations. These concepts will be used to define continuity and topological equivalence in these more general situations.

Three particularly important properties (connectedness, compactness, and completeness) are discussed in Sections 8-4, 8-5, and 8-6. The first two are topological properties; the last is not.

8-2 Metric Spaces

In Section 7-3 we mentioned four basic properties of the distance function in three-dimensional space. We shall see here that, if X is any set and d is a function which has these properties, then many of the concepts of interest in connection with three-dimensional space can be defined in the set X.

Let X be a set, and let d be a real-valued function defined for pairs of points x ∈ X, y ∈ X. The function d is a metric in X if and only if the following conditions are satisfied for all points x, y, and z of X.

The value of the function d at the points x, y [i.e., the real number d(x, y)] is called the distance from x to y. A...