Principles of Mathematics in Operations Research

10.1 Metric Spaces 139

P<...

6^

-d,—

'Oinfty •

Fig. 10.1. Example 10.1.9

Definition 10.1.10 Let (X,d) be a metric space, p 6 X, r > 0.

Br(p) = {q S X : d(p, q) < r} open ball centered at p of radius r.

Br\p] — {q £ X : d(p, q) < r} closed ball centered at p of radius r.

Example 10.1.11 X = M^2 , d = d 2. See Figure 10.2.

CLOSED BALL OPEN BALL

P

Br[P] / " Br(P)

jy s^, „'

Fig. 10.2. Example 10.1.11

Example 10.1.12 Let us have X ^ 0, and the discrete metric.

{p} , if r < 1 f {p}, if r < 1

MP) ={{p},if r = l Br\p] ={ X, if r = 1

X, if r > 1 { X, if r>l

Example 10.1.13 X = B c (a, b) = {/ : (a, b) >-> R : / is bounded}

f, 9 EX^ d(f,g) = sup{|/(s) - g(s)\ : a € (a, 6)}

Let f € X,r > 0, Br(f) is the set of all functions g whose graph lie within the
dashed envelope in Figure 10.3.

Example 10.1.14 X = W^2 with d\ metric:

d\{p,q) = 12/i -xi\ + |s/2 -x 2 \.

See Figure 10.4-