Principles of Mathematics in Operations Research

146 10 Basic Topology

E is open relative to Y •» Vp e E 3r > 0 3 Br(p) r\Y CE.

E is closed relative to Y •£> Y \ E = Y n Ec is open relative to y.

Theorem 10.1.45 LetX CY C E. Then,

(a) E is open relative to Y ^3 an open set F in X 3 E = F f]Y.

(b) E is closed relative to Y <&3 a closed set F in X 3 E — F <~)Y.

Proof. X C Y C E.

(a) (=»):

Let E be open relative to Y. Then,

Vp G E 3rp > 0 3 Brp{p) n Y C E.

Let F = {JpeE Brp(p). F is open in X.

\J[Brp(p)nY}cE FHYCE

p€E

Conversely, q G E, then

q e Brq(q) C F, qeEcY=^qeF(lY^EcFC]Y

(<=)•

E = FDY where F is open in X. Given p G E =>• p G F. Since F is open,

3r > 0 3 Br{p) C F.

Br(p)f)Y CFDY = E.

(b) (=»):
£? is closed relative to Y =>• Y \ £ is open relative to Y. Then,
3F G X open in X 9 Y\E = FC\Y.
E = Y(Y\E) = Y(FnY) = Yn(Fr\Y)c = YnFcU<D = YDFc.
Fc closed in X.
(<=) =
£ = Fn7 where F is closed in X.
Y\E = Yn{Fr\Y)c = YHFC (Fc open in X) => Y \£ is open relative
toy.
=> E is closed relative to Y. D

10.2 Compact Sets

Definition 10.2.1 Let (X, d) be a metric space, E C X be a nonempty subset
of X. An open cover of E is a collection of open sets {d : i G /} in X 3 E C
UiGi.