Question

4. Let Uαα∈A be a finite open cover of a compact metric space X. For question for (a), (b)

4. Let {U}aea be a finite open cover of a compact metric space X. (a) Show that there exists & >0 such that for each x e X, the open ball B(x; ) is contained in one of the U.'s- (b) Show that if at least one of the Ua's is a proper subset of X, then there is a largest Lebesgue number for the cover

Remark: ε is called a Lebesgue number of the cover.

(a) Show that there exists ε>0 such that for each x∈X, the open ball B(x;ε) is contained in one of the Uα’s.

(b) Show that if at least one of the Uα’s is a proper subset of X, then there is a largest Lebesgue number for the cover.

Answer 1

let {ux 4REA be a finite open cover of a melna space X. To show:- there exists exo such that for each xex the open ball B(x, E) (a)

is contained in one of the Ua's - let {uabaca be a finite open cover of our compact metna space x for every point xex select an element u (a) of fuel _so that XEU(X) Sime U(2) is open we can find some radius eso such that B(x8) Cu(2) and u(a) is an arbitrary element of aya it follows that the open ball B(x; E) is contained in one of the Uas. (b) To show: If atleast one of the va's is a proper subset of x then there is a largest lebseque number for the cover consider the collection of openballs B={BCI, 2 x6x3 - an open cover of X X - compart, there is a finite subcover B'= { B (2, lisisn} Isisn let x= mins Eɛrxi// Isian] so x>0 claim is a largest lebseque number for the cover. Suppose ESX and diam(E) <X Fix a point pee then PEB (Xi, 812) Tet a be any point in E. Then dep, a) s diam(&) < X < Ecxi) Then dr2.9) < dex.p) +depa) << + and ESB. CX;,&) therefore Ux is a proper subset of xas Then there is a largest lebseque number - of the cover