294 Chapter 5 • Continuous FunctionsComments: A countable set is of the first category, since every single
point set { x} is nowhere dense. Thus, if there are any second category sets
they must be uncountable. However, there are uncountable sets that are not
of the second category. For example, we proved in Section 3.4 that the Cantor
set is both uncountable and nowhere dense (hence, of the first category). Some
authors call a first category set meager to indicate that it is somehow "smaller"
than a second category set.
The next theorem is more powerful than it first appears. We shall put it to
work to get a very interesting corollary. You may also be interested in seeing
that Cantor's nested intervals theorem is useful in proving this theorem.
Theorem 5.7.16 (Baire Category Theorem for JR) Every proper interval
is of the second category.Proof. Suppose I is a proper interval, say (a, b) ~I~ [a, b], where a< b.
CXl
For contradiction, suppose I is of the first category. Then I = LJ An, where
n=l
each An is nowhere dense.
Since A 1 is nowhere dense, Lemma 5.7.14 guarantees that there is a proper
closed interval J 1 ~ I such that J 1 n A 1 = 0. Similarly, since A 2 is nowhere
dense, 3 proper closed interval h ~ J 1 such that J 2 n A 2 = 0. Continuing in
this way we get a sequence {Jn} of proper closed intervals such that
J1 2 h 2 · · · 2 Jn 2 · · ·and such that Vi, Ji n Ai = 0.
By Cantor's nested intervals theorem (2.5.17) 3 x 0 E JR such that
CXl
Xo En Jn.
n=l
CXl
Now, Vn E N, Xo E Jn, so xo tt An. Thus, Xo tt LJ An = I. But Xo E I,
n=l
since Vn, Xo E Jn~ I. Contradiction.
Therefore, I is of the second category. •
Lemma 5.7.17 (a) Every subset of a first category set is of the first category.
(b) Every set containing a second category set is of the second category.
(c) JR is of the second category.
(d) The union of two first category sets is a first category set. (In fact, the
union of a countable collection of first category sets is a first category set.)
(e) The set of irrational numbers is of the second category.
Proof. Exercise. •