Principles of Mathematics in Operations Research

142 10 Basic Topology

\

i

,--- P i

/ i

i i

i. /

Fig. 10.7. Example 10.1.26

Example 10.1.26 N is a neighborhood of P but it is not neighborhood of Q.

See Figure 10.7.

Definition 10.1.27 A point p S X is called a limit point (or accumulation

point) (or cluster point) of the set E C X if every neighborhood N of p

contains q of E 3 q ^ p. i.e. V neighborhood N of p, 3q € E f\ N, q ^ p.

Equivalent^, Vr > 0, 3<7 G E D Br(p) 9 q ^ p.

Example 10.1.28 £={p=(i,j/)€R^2 :Ki^2 +t/^2 <4}u{(3,0)}. Limit

points of E are all points p = (x,y) 3 1 < x^2 + y^2 < 4. See Figure 10.8.

Fig. 10.8. Example 10.1.28

Definition 10.1.29 A point p € E is called an isolated point of E if p is not

a limit point of E; i.e. 3r > 0 3 Br{p) n E = p.

Example 10.1.30 X = R, d = dx:

E=

{

i

\W~)>

0 is the only limit point of E. \fp £ E are all isolated points.

Definition 10.1.31 E is closed if every limit point of E belongs to E.

Example 10.1.32 See Figure 10.9.