Principles of Mathematics in Operations Research

10.1 Metric Spaces 145

Let (X,d) be a metric space, then

1. The union of a finite collection of open sets is open.


2. The intersection of a finite collection of open sets is open (not true for
   infinite).


3. The intersection of any collection of closed sets is closed.


4. The union of a finite collection of closed sets is closed (not necessarily true
   for infinite).


5. £ is open <=> Ec is closed.


6. E is closed <=> E = E.


7. £ is the smallest closed set containing E.


8. intE is the largest open set contained in E (i.e. if A C E and A is open
   then A C intE).

Example 10.1.41 Intersection of infinitely many open sets needs not to be
open, X = R, d(x,y) = \x - y\: Let An = (-^,n±i), n = 1,2,.... Then,
fT=i A-. = [0,1]. //0 < .x < 1 (Ami G (-£, ^) = An, Vn => a; G n~=1 A«-
Let x G 0^=1 ^n- s/tow **oi 0 < a; < 1:
If not, x < 0 or x > 1. If x > 1, 3n E N 3 1 < ^ < x, j; ^ An. Case
x < 0 is similar.

Proposition 10.1.42 Lei 0 ^ £ C R 6e bounded above. Then, sup E G £.

Proof, y = sup £, show that Vr > 0, Br(y) n L / 0: Since y — r<y=>y — r
is not upper bound of E. Bx G £ 9 y >_ x > y — r =$• x € (y — r,y --r)C\E =>
Br{y)C\E^%. 0

Let (X, rf) be a metric space and I / 7 C X, then 7 is a metric space
in its own right with the same distance function d. In this case, (Y, d) is a
subspace of (X, d).
If £ C Y, E may be open in (Y, d) but not open in (X, d).

Example 10.1.43 X = R^2 , Y = R, £ = (a, 6): WTierc considered in R, £ is
open whereas E is not open in R^2 , as seen in Figure 10.10.

•** ~ •%

Hr^m

a N\ c-^// b

Fig. 10.10. Example 10.1.43

Definition 10.1.44 Let E C Y C X. We say E is open (respectively closed)
relative to Y if E is open (respectively closed) as a subset of the metric space
(Y,d).