Proposition 5.1.4: Characterizing Open Sets
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.
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).