Elements of Metric Spaces — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

(c) Let (X,d) a metric spac.'e and AEX. Define f: XiR as f(x.) = d(x,A) = inf{d(x,y): y € A}, x€x Prove that f is uniformly continuous over Xj. 5. 6. (a) Show that the metric spaces (X,d) ,.and (X, p), where : p(x,y,-#x(#;vx,y€x are equivalent. (b) Let (X, d) be a disconnected metric space. Prove that there exists a continuous mapping of (X, d) onto the .discrete two element space (Xo, .qo). (c) .Define a disconnected metric space. Give examples of a disconnected. and a connected metric space. (a.) Let Y be a subset of a metric space (X,d). If (Y, dv) is compact then prove that Y is a closed subset of (X, d). ®) Let i be a continuous function .from a compact metrig space (X, dx) into a metric spac? (Y, d¥)t Then show that the range f(X) of f is also (c) Give examples of infini;e subsets of R vyhich .are : (i) Both connected and compact (ii) N`either connected nor compact` (iii) Co`mpabt but not connected (iv) Connected but not compact Narie of the Paper Name of. the Course : Elements of Metric Spaces 1.. .(a.) Let (X, d) be a metric space. Define the function d': X x X i R by d,.(x,y)- d(x,y) Show that d' is a thetric on X. (b) Define the conver;ence of a Sequence .ip a metric space (X, d).. Prove that the limit of ,a sequenc? is (c,i Find open and closed ba~lls in the metric .space (R2, d[) where di(X!y)F`|Xi-yi|+|X2-y2|9X=(Xi,X2),y=(yi,y2)€R2, with centre (OzO) and radius 1. Sketch it al`sb`. 2. (a) Define an open subset in, a metric space (X, d), Show that every open ball in (X, d) is an open set. Is the converse true? Jirstify. (b) For an`y two sub.sets A, a of a metric spa'ce (X, d). Prove that (ii) A0 u 8° c= (A + 8)a. Give an example to / show that the equality maynothQld in this (c) For any two subsets A, a of a metri.c space (X, d). (ii) Z=i5 c= A r`B . Give an example to show that equality may not hold in this cas`e, 3. (a) Let (X, d) be metric space where X = R and d is usual metric. Let Y = [0,1) be a subspace of X. Show that F] = [0,0.2] and F2 = [0.8,1) are closed inY. (b) Let (X, d) a metric S`pace and Y b-e a sub;pace X.-.1 Let z e Y and r > 0 then (i) Show `that S¥(z, r) = Sx(z, r) fl Y. (ii) Show that, §y.(z,r) = Yr`Sx(z,r). (c).Prove that a gomplete subspace of any metric space ,is glosed; Is converse true? Justify. (a) Let (X, dx) an¢ (Y, qy) be two metric spaces and A E X then„ prove. that a function f: A i Y is continuous.``at. a € .A. if wihenever a sequence {Xn} in A converges to '`a' then the sequence (b) Let (X, dx) ,arid (Y, qy) b`e two metric spaces. A mapping f: .X + I is continuous if and only if f-I(F) is closed in X-for all cl,a,sed subsets F of Y.