5.1. Open and Closed Sets

Proposition 5.1.4: Characterizing Open Sets

Let U R be an arbitrary open set. Then there are countably many pairwise disjoint open intervals U n such that U = U n


This proposition is rather interesting, giving a complete description of any possible open set in the real line. To prove it, we will make use of equivalence relations and classes again. First, let us define a relation on U: Is this relation indeed an equivalence relation ? Assuming that it is, we know immediately that U equals the union of the equivalence classes, and the equivalence classes are pairwise disjoint. Denote those equivalent classes by U n

Each U n is an interval: take any two points a and b in U n. Being in the same equivalence classes, a and b must be related. But then the whole line segment between a and b is contained in U n as well. Since a and b were arbitrary, U n is indeed an interval.

Each U n is open: take any x U n. Then x U, and since U is open, there exists an > 0 such that ( x - , x + ) is contained in U. But clearly each point in that interval is related to x, hence this neighborhood is contained in U n, proving that U n is open.

There are only countably many U n: This seems the hard part. But, each U n must contain at least one different rational number. Why ? Since there are only countably many rational numbers, there can only be countably many of the U n's (since they are disjoint).

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