Principles of Mathematics in Operations Research

1 < \{(3T - 2 - aT + 1] = ^ ")3"

10.4 Connected Sets 151

1 a3~"3 1

n - 1 a

— > ^ > — > 1,

is what we want. So, a > 4. Then, (^f^^1 , ^|^) C (a,/?) C C, Contradiction!

•

Proof (Property 4). Let x £ C be an arbitrary point of C. Let Br(x) =

(x — r,x + r) be any open ball centered at x. Find n 6 N 3 4r <

r, x G C = CC=1Em => x e £„ = /f U ••• U /£,, (disjoint intervals).

So x £ /" for some j = 1,2,... ,2™. Then, x £ (x - r,x + r) D /j^1 and

length(/™) = -L < r => /" C (x - r, x + r).

Let y be the end point of 7™ 3 y ^ x. Then, y £ C n(x — r,x + r) => x is

a limit point of C. D

10.4 Connected Sets

Definition 10.4.1 Let (X,d) be a metric space and A,B C X. We say A

and B are separated if A n B = 0 and .4 n B = 0. /I subset E of X is said to

be disconnected if 3 two nonempty separated sets A,B3E = AUB. E C X

is called, connected if it is not a union of two nonempty separated sets, i.e. 3

no nonempty separated subsets A, B B E — AL) B (V A,B pairs).

Example 10.4.2 X =• K^2 , with d 2 ,dx or

Let E = {(x,y) : x^2 < y^2 } = {{x,y) : \x\ < \y\}. See Figure 10.11.

CONNECTED DISCONNECTED

Fig. 10.11. Example 10.4.2

Theorem 10.4.3 A subset E ^ 0 of R is connected if and only if E is an
interval (E is an interval if and only if z,x £ E and x < z => Vy with x <
y < z => y £ E).