1550251515-Classical_Complex_Analysis__Gonzalez_

Topology of Plane Sets of Points 111

metric

00

d(x, x'). = [2)xn - x~)^2 ]

1

f2

1

and let M be the subset

M = {em: em = { 8~} ;:1 }

where s:i, = 0 if l i= m, s:i, = 1 if l = m (Kronecker's deltas); i.e., M

is the set of the "units" ei = {1,0,0, ... }, e 2 = {0,1,0, ... }, and so on.

Let 0 = {O, 0, 0, ... }. Then d(O, em) = 1, so M is bounded since it can

be enclosed in a ball with center at 0 and radius 2. However, M is not

totally bounded since

( i i= j)

and M cannot be covered by finitely many neighborhoods N 6 ( em) with

c = %./2.

Definition 2.38 A metric space (S, d) is said to have the Bolzano-
Weierstrass property if every infinite subset of S has a limit point in

s.

As the reader may recall, the classical Balzano-Weierstrass theorem

refers to a closed and bounded (i.e., compact) subset of the real line, and

it is stated as follows: If K is a closed and bounded subset of JIV, then

every infinite subset 6f K h~s a limit point in I<. A gen~ral metric space

may or may-not possess the Balzano-Weierstrass property. For instance,

(C, d) does not have the property since the infinite set [i, 2i, 3i, ... } has no

limit poirit in C. · -·
We wish to show that in metric spaces the Balzano-Weierstrass property
is equivalent to the Heine-Borel property, i.e., fo compactness. To this
end we first introduce the notion of sequentially compact space, which is
closely related (in fact, equivalent) to that of space possessing the Bolzano-
Weierstrass property.

Definition 2.39 A sequence { Xn} ~ in a metric space ( S, d) is said to
converge to a point x 0 E S, or to have x 0 as a limit, iff for every c > 0
there is a positive integer N 6 such that n ~ Ne implies that ~

d(xn, xo) < c

Alternatively, {xn}~ converges to xo iff Xn E N 6 (xo) for all values of n

equal to or greater than a certain positive integer N (depending on s > 0).

Jt follows that if both m and n are ~ Ne, then

d(xm,Xn) ::S d(xm',xo) + d(xo,xh) < 2s