Principles of Mathematics in Operations Research

10.1 Metric Spaces 143

CLOSED OPEN Not CLOSED Not OPEN

(o) (d) (o)

Fig. 10.9. Example 10.1.32

Definition 10.1.33 E is perfect if it is closed and every point of E is a limit

point of E; i.e. if E is closed and has no isolated points. E is bounded if

3M > 0 9 Vp, q G E d\p, q] < M. E is dense in X if every point of X is

either a point of E or a limit point of E.

Example 10.1.34 X = K, E — N is unbounded. Suppose it is bounded.

Then, 3M > 0 B Vx, y G N, \x - y\ < M. Let n G N be B n > M + 1 =>

|l-n|=ra-l<M-»n<M + l. Contradiction!

Example 10.1.35 X = M, E = Q (Q is dense in R; i.e. given x G K either

x G Q or x is a limit point ofQ). Let i£l, if x G Q, we are done. If x ^ Q,

we will show that x is a limit point of Q:

Given r > 0, Br{x) = (x — r, x + r). Then, 3y € Q 3 x — r<y<x + r=>

y G Br(x) n Q and y ^x => x €R, y G Q.

Let us introduce the following notation:

E': set of all limit points of E.

E — E U E', E is called the closure of E.

pE E <=* W > 0, Br(p) n £ / 0.

Proposition 10.1.36 Every open ball Br(p) is an open set.

Proof. Let q G Br(p), we will show that 3s > 0 3 Bs(q) C Br(p):
q G Br(p) => d(q,p) < r, let s = r - d(q,p) > 0. Let z G Bs(q),

d(z,p) < d{z,q) + d{q,p) < s + d(q,p) =r =>• z G Br(p). •

Theorem 10.1.37 p is a limit point of E if and only if every neighborhood
N of p contains infinitely many points of E.

Proof. (<=): trivial.

(=>): Let p be the limit point of E. Let N be an arbitrary neighborhood
of p. Then, 3r > 0 B Br(p) C N. Since Br(p) is a neighborhood of p