Principles of Mathematics in Operations Research

10.1 Metric Spaces 141

Example 10.1.19 See Figure 10.5. q G intE but p g" intE.

Fig. 10.5. Example 10.1.19

Example 10.1.20 Let X be any set with at least two elements, with the dis-

crete metric:

Letpe X, E = {p}. Then,

intE = E, r < 1 => Br(p) = p c E => p £ intE.

Example 10.1.21 Let X = R^2 with d 2 metric. See Figure 10.6.

T

Fig. 10.6. Example 10.1.21

E = {p = (x,y) £ R^2 : 1 < x^2 + y^2 < 4} =>

intE = {p = (x,y) e M^2 : 1 < x^2 + y^2 < 4}.

Definition 10.1.22 E is said to be open set if intE = E, i.e.

\fpeE,3r>0 3 Br(p) C E.

Example 10.1.23 In K^2 , E = {p = (x,y) £ E^2 : 1 < x^2 + y^2 < 4} is open.

Remark 10.1.24 By convention, E = 0, E = X are open sets.

Definition 10.1.25 Let p £ X. A subset N of X is called a neighborhood of
p if p G intN.