1550251515-Classical_Complex_Analysis__Gonzalez_

Topology of Plane Sets of Points 113

that d(xa,xi) 2: c and d(x 3 ,x 2 ) 2: c. Proceeding in the same manner,

we obtain either a finite covering of S by c-neighborhoods or an infinite

sequence {xn} such that d(xi,Xj) 2: e for i f=. j. This last case cannot

occur since then neither the sequence {xn} nor any of its subsequences

would converge, which contradicts the hypothesis that (S, d) is sequentially

compact.

Theorem 2.23 If the metric space (S, d) is sequentially compact, then

it is compact.

Proof Let LJ G(){ be an open covering of S. Since (S, d) is sequentially com-
pact, by Theorem 2.22 there is a finite covering of S by s-neighborhoods,
i.e.,
n
S = LJ Ne(xi)
i=l
Choose c = 1/k (k a positive integer) and suppose that it is impossible to
obtain a finite subcoverin~ of S from { G (){}. Then for each k there is at least
one neighborhood Ne(x): ) which cannot be' covered by a finite number of

the sets G(){. Consider one such neighborhood for each k and let_ { x):)} be

the sequence of its centers. Because (S, d) is sequentially compact, there is

a point p E S which is the -limit of a subsequence of { x)~) }. Let Go be a

set of the family { G (){} containing p. Since G 0 is open, there exists 5 > 0
(kJ.

such that N 0 (p) C G (^0) ' • Next, choose. k and a center xi· k so that l/k <^1 / 25
and d(p,x~:)) <^1 / 2 5. Then it follows that
N1;k(x):)) C Go
so the neighborhood N 1 ;k(;;:)) is covered by a single set Go. This is a
contradiction.
Thus we have shown that in metric spaces the Heine-Borel property
implies the Bolza~o-Weierstrass property, that the latter is equivalent to
sequential compactness, and finally, that sequential compactness implies
compactness (the Heine-Borel property). Hence all three properties are
equivalent in the case of a metric sp·ace. The result holds for any compact
subset of a metric space with the relative metric.
Theorem 2.24 Let {Fn}~ be a sequence of nonempty closed subsets of
a compact metric space (S, d) such that Fn :J Fn+l · Then the intersection
F = n::"=i Fn is not empty; i.e., there is at least one point that belongs to
each Fn (Cantor's lemma, also known as the closed nested sets property).