Introduction to Real Analysis by Michael J. Schramm

Author:Michael J. Schramm
Language: eng
Format: epub
Publisher: Dover Publications
Published: 1996-06-26T16:00:00+00:00

11.2 THE COVERING PROPERTY

The converse of Theorem 11.4 would be very useful. Deciding whether a set is closed and bounded would seem far easier than deciding whether it is compact. Our solution to this problem will lead us into some very abstract mathematics. Being abstract often means looking at a problem topologically, that is, finding a way to express an idea in terms of open sets. The definition of compactness is only partly topological. It involves the (topological) issue of continuous functions, but we must also consider the (nontopological) question of the ordering of the real line to make sense of "maximums."

The Covering property is a tricky concept. To get an idea what it is about, let us consider two sets we know to be compact (a finite set and the set S in Example 11.1.5) and see if they have anything else in common. Our claim that "Every function whose domain is a finite set is continuous" is based on bits and pieces of other proofs. If we were to assemble a detailed proof of this statement, we would see that it hinges on the fact that we can enclose a finite set in a collection of open intervals. (That every function on such a set is continuous then follows because each point of the set is open.)

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