“real: chapter_04” — 2011/5/22 — 23:17 — page 16 — #16
4-16 Real Analysis
points of G, then I
x
and I
y
are either disjoint or identical. Indeed, if they
have a common point z, then I
x
= I
z
and I
y
= I
z
and hence I
x
= I
y
.
Consider the class I of all distinct sets of the form I
x
for points x in G
(note that I may be a finite of infinite collection). This is a disjoint class
of open intervals and G is obviously its union. It remains to prove that
I is atmost countable. Let G
r
be the set of rational points in G. G
r
is
clearly non-empty. We define a mapping f of G
r
onto I as follows: for
each s in G
r
, let f (s) be that unique interval in I , which contains r. G
r
is countable and hence I is atmost countable (Note that f is onto and
from Theorem 2.5.4(ii) we can deduce ...