1550251515-Classical_Complex_Analysis__Gonzalez_

118 Chapter 2

Theorem 2.28 (Cantor) A metric space '(s, d) is complete iff for any

sequence {Fn} of nested nonempty closed sets with 6.(Fn) -+ 0, the

intersection n:=i Fn consists· of a single point.

This is analogous to Theorem 2.24 plus its corollary. The reason for this

is contained in the following property.

Theorem 2.29 Every compact metric space is complete.

Proof In fact, if the metric space is compact, we can use Theorem 2.24,

together with its corollary, with the Fn replaced by closed spheres En.

Then, by Theorem 2.27, it follows that the metric space is complete.

Clearly, the converse is not necessarily true. For example, <C is complete

yet noncompact.

Definition 2.43 A subset ]( of a metric space (S, d) is said to be of the

first category if it is the union of a countable collection of nowhere dense

sets. Otherwise, J( is said to be of the second category.

f( Theorem 2.30 Every complete metric space is of the second category ..

Proof Assume that there is· a complete metric space S which is of the first

category, i.e., such that S =· LJ:=i An, the An being nowhere dense sets

in S. Then the set Ext An is everywhere dense in S, so that Ext An= S. It

follows that any point x E S is a contact point of Ext An or, equivalently,

that every spherical neighborhood N6(x) contains a point b E Ext An. Since

Ext An is an open set, there exists a sphere N 0 1(b) C N6(x) and contained

entirely in Ext An- Choosing 0 < r < 8', we get a closed sphere E[b, r]

contained in N 6 (x) but containing no points of An.

Thus we can find a closed sphere Ei of positive radius ri < 1 that does

not intersect Ai. In the interior of this sphere we can find a dosed sphere

E 2 of positive radius r 2 < t that does not intersect A 2 , and so on. The

sequence {En} of the nested spheres is such that for each n, En does not

intersect any of the sets Ai, A 2 , ••• , An, and rn -+ 0 as n -+ oo. Hence,

by Theorem 2.27, we have

00

i.e., the :intersection of the spheres is a single point y E S. By the con-

struction of the En the pointy does not belong to any of the sets An. This

is im_possible since we have assumed that S = LJ:=i An. Therefore, S is

of the second category.

If the metric space (S, d) is not complete, it is always possible to embed

S in a complete metric space .. .S*.