Fundamentals of Topology — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Your Roll Nd............... AII questions cany equal marks. (a) Let (X, d) be a metric space and let B(x, r,) and B(y, rr) be two intersecting balls containing a common point z. Show that there exists an r, > 0 such that B().-irr) c B(x, rr) . B(x, rr). Let (X, d) be a discrete metric space then show that xD -) x in (X, d) if and only if xo = x for all n except possibly for finitely many n. (b) Let X = IR. with the metric d-(x, y) : max {lx, y,l: I < j < .r). Then, show that the convergence of sequences in (1R', d-) is equivalent to coordinate wise convergence. (c) Define the notion of base for the open sets in a metric space. Show' that there exists a base for open sets in a metric space. Further, define the concepl of isometric melric spaces. (a) What are locally connected metric spaces. Is the property of being locally connected preserved by continuous maps? Justify. Give example of a metric space that is connected but not locally connected. (b) Let E be a closed subsei of a metric space (X, d). Prove that E is nowhere dense iff for every open subset G there is a ball contained in G - E. Show that every non empty open interval of lR is of calegory II. (c) State Tietze extension theorem. Show that the function cannot be extended to a continuous function on lR, but the function nl*)-sln^- xe lR lor . ,'. "" l"l can be extended to a continuous function on lR. (a) Show that a subset of a totally bounded metric space is totally bounded. Is the metric space (i, d) totally bounded? Justify your answer. (b) Let X denote the set of all bounded sequences of real numbers with metric d(x,y) : sup {lx,-y,l: i > l}. Then, prove that (X, d) is inseparable. (c) Let X be the set of all complex numbers and d: X x X -+ lR be defined by d(zp zr) = 0 when z, : z, and d(2, zr) : lz,l + lzrl when z.t + 22. Show that (X, d) is a metric space which is complete. (a) Is the real line IR, with the standard topology compact? Is the following subspace of R. compact? Justify your answers. (b) Prove that the union of a collection of connected subspaces of X that have a point in common is connected. (c) Define a continuous function from a space X to space Y. Prove that for continuous functions f : IR -+ IR, the e - 6 definition of continuity implies the open set definition of continuity. (a) Show that B : {(a,b): a,b e Q, a<b} is a basis that generates the standard topology 5. on R. Also, show that the collection generates a topology on IR dilTerent from the lower limit topology on R. (b) Let fr: A -r X and f,: A > Y be any two maps. Detlne f: A -+ X x Y by the equation f(a; = (f,(a). fr(a)) VaeA. Prove that f is continuous if and only if f, and f, are conlinuous. (c) Show that a space X is connected if and only if the only subsets of X that are both open and closed are $ and X. (a) Let A be a connected subspace of X. If A c B g A, then B is also connected. (b) Prove that the image of a compact space under a continuous map is compact.