58 A anndastie te Metric SpacesZR. BChapter-1: Motrie SpacesShow that each open sphere is an open set in a metric space.Neighbourhoods 59Short QuestionsSolve / write answers of the following short questions:State the Cauchy Schwarz inequality.Q.5Q.2PU, 2013: 2012: 2011: 2009: S2002, 2002, 1997: 1990: 1993: 1992 (BAU.Se.()PU, 2012 (US MathPU, 12 (BSShow that a subset of metric space is open if and only if it is theunion of open spheres.Q.6What will be the open and closed spheres each of radius 2.(ii)discrete metric space X? Whether these are same or differennDefine discroto metric space.Show that every nonempty subset of a discrete metric spacePU, S2002: 1993 (B.A/B.Sc.EProve that the set A= ((x.y)eR?: x +y? A metric d on X is said to be bounded metric on X if there exists ametric space.positive number M such that d(x.y)SM for every pair of points xPU, 1994: S1993 (BABand y of X.(vi)Let (X,d) be a metric space. If U,.U, are two open sets in x > Let (X,dy be a metric space. A subset of X consisting of all thosethen prove that U, nu, is also an open set.PU, 2000 (BAES(vii) Let X be a metric space and let (x.} be a singleton subset of Xpaints of X whose distances from some fixed point of X are less thansome fixed positive real number is said to be an open ball or opensphere.> An open ball in a real line Ris an open interval.then show that X-(x.) is open.PU, 2007: 1994; S1993 BAE > An open sphere of radius O The open sphere of radius greater than one in a discrete metricspace is the whole space.PU, 2001 (BABS> Let (X,d) be a metric space. A subset of X consisting of all thosepoints of X whose distances from some fixed point of X are lesser orequal to some fixed nonnegative real number is said to be a closedball or closed sphere.> A closed ball in a real line Ris a closed interval.> Let (X,d) be a metric space. A subset U of X is called an open set iffor every x in U, there exists some positive real number r such thatLong QuestionsQ.3Show thatXq, Xg, Y4,Y2 €R is a metric on R?.PU, 2009 (BABQ4Let x, y be to points of R" or C", then show that> In a metric space X, the empty set and the full space X are open11/211/2sets.Open set in Ris an open interval.PU, 2009: 2010; 2011 (BAB S* Every nonempty subset of a discrete metric space is open,

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Chap Neighbourhoods 59 Short Questions Solve / write answers of the following short questions: as Show that each open sphere is an open set in a metric space. PU, 2013. 201 201E0 S2002, 2002 1997, 1 93, T2 A Q.2 State the Cauchy Schwarz inequality. PU. 2012 S Math a6 Show that a subset of metric space is open if and only if it is the union of open spheres. What will be the open and closed spheres each of radius 2 discrete metric space X? Whether these are same or difereo PU. S2002 11BAB.S a7 Prove that the set A- {{x.y) « R°: x² +y? A metric d on X is said to be bounded metric on X if there exists a metric space. positive number M such that d(x.y)sM tor every pair of points and y of X. (v) Let (X,d) be a metric space, If U,Uz are two open sets in x > Let (X,0) be a metric space. A subset of X consisting of all those then prove that U, nUz is also an open set. PU2000BA (vil) Let X be a metric space and let (x.) be a singleton subset of X points of X whose distances from some fixed point of X are less than some foxed positive real number is said to be an open ball or open sphere. then show that X-(x.) is open. > An open ball in a real ine Ris an open interval. PU, 2007. 1994; 5193 0 AL > An open sphore of radius 0 The open sphere of radius greater than one in a discrete metric space is the whole space. PU, 2001 BAL > Let (X,d) be a metric space. A subset of X consisting of all those points of X whose distances from some foxed point of X are lesser or equal to some fixed nonnegative real number is said to be a closed ball or closed sphere. > A dosed ball in a real line Ris a closed interval. > Let (X,d) be a metric space. A subset U of X is called an open set if for every x in U, there exists some positive real number r such that Long Questions a3 Show that *.2.YY2 eR is a metric on 9. PU. 2009 (A Let x, y be to points of or C, then show h Q4 > In a metric space x, the empty set and the ful space X are open sets. Open set in Ris an open interval. PU, 2009 2010, 2011 (BAS * Every nonempty subset of a discrete metric space is open.