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
Context

Proof:

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:

if a and b are in U, we say that a ~ b if the whole line segment between a and b is also contained in 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).

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 26, 2007