Interactive Real Analysis
- part of Proposition 5.1.4: Characterizing Open Sets
R be
an arbitrary open set. Then there are countably many pairwise
disjoint open intervals
U n such that
U =
U n
ContextProof:
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).
