Question

Problem Set 7 Functions between Metric Spaces 1. Construct a metric on (0,1) to make ((0.1), d) a complete metric space.

Answer 1

if ty Soletion cet x5 1011). for diye (011) define dix,y)= so 22 if sty. (ro, 1= x,d) is metric space. 0 Cet my ex = 10,1) by definition of d drony) >0 & noyex=101) © By definition of d. diniy)=0 iff nay. ☺ cel myex=101)) - 2F Ray =) dray)0dly,ze) if zety 7 diniy )= 1=dly, e diary)= deyirler & dryfX=(0,1) » cet 2019, 26 X=(0,1) af ecy » dialy) 20 Ld12,8) + drziy) A dlary) < d (012) t dezny)

af xay =) diu,y)= Either 212 1. or y72 ordey, z) = 1 din 12) an = druy) = 0(4,2)+d120) (x,d) is metric space Now . - 1 we show that it is complete metic space at {sen} a be a cauchy sequence in t. By defa of cavery po to be there exist a positive integer such that - dizen, sms) </ * nima, N. - a desenizem)=0 a ena rem & n,m>, . after The sequence certain stage, {en} becomes constant & every constant sequence is convergent {zen} is convergent = 27 xacoil) is complete memic.