Principles of Mathematics in Operations Research

152 10 Basic Topology

Proof. Let us mark the statement

2, x £ E and x < z => Vy with x<y<z=$-y€E (*).

Let E ^ 0 be connected. If £" is not an interval => (*) does not hold. i.e.

3x,z G E 3 x < z and 3y 3 x < y < z and y ^ E. Let yly — (—oo,y) n

U, i?w = (y, +oo)niJ. i4y 7^ 0 (because x G J4W) and By^% (because 2 G 5y).

Ay U By = [(-oo, y) U (?y, oo)] (IE = E. Ay C (-oo, y) =^> .4y C (-oo, y] and

5,; C (y, oo) => Ay n Sy C (-oo, y] n (;</, + oo) = 0 => Ay n 5y = 0.

Similarly, Ay II By = Q => E is disconnected, Contradiction!

Suppose ZJ is disconnected. Then 3 nonempty separated sets A, B 3 A U B =

E^1. Let i e A,t/ e B. Assume without loss of generality x < y (because

A n B = 0,_z ^_y). Let 2 = sup(A n [x, y}), then z E An [x, y\ C A (because

Ac B => A C B), z ^ B. Since x £ An[x, y], we have x < 2. 2 G .4 n [x, y] C

[x, y] = [x,y] =$• z < y =$• x < z < y.

z = i] G A 1 -

If ' > => jy G J4 n B = 0, Contradiction; hence, 2 < iy.

So, x < z < y, and 2 £ A.

If z £ A =$• x < z < y. So x,y €. E 3 x < y and Z3I<2<J, z ^ E because

z <£ B,z g A. So (*) does not hold.

lfz€zA=>ztf:B (because sets are separated).

Claim: (z,y) <£ B. If not, (z,y) C B4 (z,i/) C B 4 [z,i/] C B =* zeB,

Contradiction.

Therefore, 3z\ G (z,y) 3 z\ & B => x < z < z\ < y => z\ € [x,y].

If 2] G A, then z\ < z => z\ ^ A, z\ tf: E. =$• x,y €• E 3 x < y and

2i 3 x < z\ < y, Contradiction to (*)! •

Problems

10.1. Let X y^ 0 be any set. Let d, g be two metrics on X. We say the metrics

d and y are equivalent if there are two constants:

A, B > 0 3 Ag(p, q) < d{p, q) < Bg(p, q), Vp, qeX.

Show that the metrics di,d 2 ,d 00 for Rk are all equivalent, i.e. find A,B.

10.2. Let (X,d) be a metric space, p G X, r > 0. One is inclined to believe
that£?r(p) = Br\p\ i.e. the closure of the open ball is the closed ball. Give an
example to show that this is not necessarily true.

10.3. Show that a metric space (X, d) is disconnected if and only if X has a

nonempty proper subset which is both open and closed.

10.4. Consider the Printed Circuit Board (PCB) given in Figure 10.12 having

36 legs separated uniformly along the sides of the wafer. Suppose that a CNC