Question

1. Her x,y e X için e(x, y) = min{1. d(x,y)} şeklinde tanımlanan e met- riğinin d metriğine denk olduğunu gösteriniz.

Show that for each x, y ∈ X, the metric e defined as e (x, y) = min {1, d (x, y)} is equivalent to the d metric.

Answer 1

given i (x,d) be metoic space, The metric é defined e (2,4) = min LI d 186,4)) 18 is equivcelent to the d metric Colution - fost g will show is metro e1x,4)= min {!, d(8,4)} Краер ☺ d (x,y)>,0 > min $1,218,4)} », elx,y)>0 (10) e(x, y) = = 0 - min {1, d (2,4)} = 0 d CM, 4) = 0 essay nüs e(x,4) = min S1, 0(2,4)} = min !, d 14,4) €14,x) elx, 4) v X, Y EX let x, y, I be live any thoer point of X, If at least one of d 146,y) dly, z) is greateo than equall & ad (8,9)>, I to one then mins!, d (8,413 =1 so thool elx,y)= €18,4) + 219,12)> I > RIH, Z)

dly,z)</ Also case d (x, y)<1d when we have ela,y) = min {d,&,4), 1} = dia,y) Ply, )mins 214,7),1] =d14,2) P(x, y) + fly, z) d144) +d14,2) >, d 18,2) by tringle intequality > e(, 2) tla, z) <1042) elx,y) +214,2) > el X, Z) x x,y,z EX 4 Their t e is metric on X Hence syt X Now eesti 9 we'll show é is ecleivalent d Leer meist show that every openset (x, e) in open in (x,d) and vise- versa For this of x in in (*, e) and let Ey be any open subxet REG then Ţ openset f yex: f(x,y)<8} .CG -@ Play) - min 1,0 (4,4) e(t, y) adla, 4) Ex Y EX Hence using , we get şi yeX!d(8,47203 € sy ex : e(x,y)<2} C G showing that each point of G is the centre of open xphare in ix,d) contained in a

Hence G is also open in (x,d). Hence enery open set in (x,e) is open in (X,d) Next ut H be any open set in X,d) Then for each x € H 3 open sphase {ye x: 018,4)<83c+ r'= min{r} so that rlsr Then let lece halo { yox.2(4,4)<r 3c {yex:d1441735 showing that each point of H is the contre of open sphare contained in in lx, e) Hence H is also open in (x, e) Hence every openset in (x,d) is open in (x, e) are equivalent metric / Henco d & e