Principles of Mathematics in Operations Research

140 10 Basic Topology

Fig. 10.3. Example 10.1.13

Example 10.1.15 X = K^2 with rfoo metric:

doo(p,q) = max{|t/i -xi\,\y 2 - x 2 \} •

Definition 10.1.16 A subset E 7^ 0 of a vector space V is convex if

tp + (1 - t)q G E whenever p,q G E and t € [0,1].

Proposition 10.1.17 X — M.k withd 2 , d\ or d^ metric. Then, every (open)
ball Br(p) is convex.
Proof. Using c^ metric:
Fix Br{p). Let u,v £ Br(p), 0 < t < 1. Show that tu + (1 - t)v G Br(p) :
Letp = (pi,...,Pk), u = (ui,...,uk), v = (vi,...,vk). Then,
doo(tu+ (1 - t)v,p) = d 00 (tu + (1 - t)v,tp+ (1 - t)p)

= max{|<Mi + (1 - t)vi - tpi - (1 - t)pi}i=1
= \tuj + (1 - t)vj - tpj - (1 - *)pj-| = \t{uj - pj) + (1 - i)(u,- - p 7 -)l
< |<||uj-pj| + |l-t||uJ--pj| = td 00 (u,p) + (l-t)d 00 (u,p) < tr+(l-t)r = r.
D

Definition 10.1.18 Let (X,d) be a metric space, E C X. A point p G E
is called an interior point of E if 3r > 0 3 Br(p) C E. The set of all
interior points of E is denoted by intE or E° and is called the interior of E
(intEcE).

(^1) / X
1 / S
1/ /
Y %.
K
iIvs X X
(^1) \
X ^
X N
X \'
r Xl
Rectilinear
I / \ >
P r \
Euclidean Tchebycheff's
Fig. 10.4. Example 10.1.14