Now, we observe f -1 (V) n Ui = f,--1 (V) n U. Use Baire's theorem to show that R is uncountable. Ex. Then yn -4 x. x E X and d(x n ,y n ) (b) Let x n O. Can you interpret f (x) as a geometric relation between x and the interval J? Draw pictures.) Ex. Assume the contrary. (We wanted to give precise E.- N argument. Example 5.2.23 (Topologist's Sine Curve - I). The next example proves something stronger.) a topology T on X. We shall give explicit estimate for II In — fmll i to quell your doubts, if any. 4.2.6. 3.3.8. Thus the given family is an open cover of (0,1). COMPACTNESS 100 Proof It suffices to show that a given norm on Rn is equivalent to the Euclidean norm. Thus the claim is proved. The converse is not true. See Definition 2.1.5.) (4) Every sequence has a convergent subsequence. Let U := R \ Z. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. An immediate, though trivial, application is Ex. 3.3.3. Definition 1.2.66 (Equivalent Metrics). CONTINUITY 72 3.3 Topological Property Definition 3.3.1. The reader is encouraged to supply the details. Show that any linear map T: Rn -> X is continuous. TOPOLOGY: NOTES AND PROBLEMS Abstract. 1.2.64. 0, then there (iii) If f : X RI is continuous at x E X and if f (x) exists r > 0 such that f(x') 0 for all x' E B(x,r) and the function g(x') := 11 f (xl) from B(x,r) to R is continuous at x. Analogous results hold if we replace R by C. Proof Let x i, x. Since X is complete, (xn ) converges to some point x E X. Show that any contraction is Lipschitz continuous and hence it is uniformly continuous. We call this path -y a linear path. We want a continuous function that vanishes on A. Find the distance d(A, B) between A and B where (a) A = Q and B = R\ Q. Thus, Ac B(xk,2rk) C U2 (ik ). Ex. We say that a subset A C X is an E-net if dA(x) < E for any x E X. Proof Since this is a very useful result, we shall give a proof. Any two open balls in 1Rn are homeomorphic. Let uous. By Archimedean property of R, we can find n E N such that 1/n < 1 — E. Let N := max{m, n } . 132 CHAPTER 6. Show that a map f : X —> Y is continuous if for every closed set V c Y, its inverse image f -1 (V) is closed in X. Ex. (This is the standard definition of the boundary of a set.) This along with the fact that c E U allows us to conclude that [a, cl C U and hence c G E. We now show that c = b. Hint: Think of a family of open balls indexed by x E U. f --1 (C) = f --1 (C) n X = f -1 (C) n (uiAi ) = ui (fi--1 (C)n Ai ) This completes the proof. For, if k < 1, oo 1-1 d(x nk , dni ) oc as seen earlier. I would like to receive suggestions for improvement, corrections and critical reviews at kumarrsaOsankhya.mu.ac.in Mumbai Hint: N and In + : n E NI. The reader should prove the theorem on his own. Show that R2 \ {0 } is connected. Show that d is a metric on X. We say that f is continuous at x if given an open set V 3 f (x), we can find an open set U 3 x such that f (U) C V. Ex. (Draw a picture.) Let X, Y be metric spaces. 3.4.7. Show that a subset D c X is dense if it is E-net for every E > 0. EQUIVALENT DEFINITIONS OF CONTINUITY 65 by the same set I and with the property that ai < bi for each i G I. {s r, } so that sr, — srn < — s n, for m < n. Eic'c' 6.1. Ex. Show that the function fx defined by fx (y) := d(x, y) is continuous. 'We used the fact that 2 -k 0. (This approach is the standard one!) 1.2.30. Compare this result with Theorem 6.1.3 - b - a, the length of the If J = [a, is an interval, we let f(J) interval. Show that the set of all invertible matrices in M(2, R) is open. 3.2.11. Hint: Let U be open in 100 and x E U. be an NLS. Since g is continuous at y, it follows that g(yn ) ---* g(y) or what is the same, g ( f (x n )) --- g(f (x)). We additionally find the money for variant types and with type of the books to browse. A nonempty open set in JR is the union of countable family of pairwise disjoint open intervals. Assume such an 8 exists. 4.1.14. We define f (x) := g(x) if x E A infIg(a)d(a,x):aeill if x A. dA(x) Note that since 0 < g(a) < 2 for a E A, we have 1 < f(x) < 2 for x E X \ A. Let f : X ---4 X be such that d(f (x), f (y)) < d(x, y) for all x, y e X with x y. (3) the inner product map Rn x Rn -- TR given by (x, y) i-- (x, y). Let (X, d) be an unbounded connected metric space. So, if 6 > 0 is given, then we may choose 6 = E. On the other hand, look at g: (0,1) —> 11/ given by g(x) =11x. 3.2.12. Ex. Let d be the Euclidean metric on R2 . Consider X := {(x,sin(14)) : x > 0} U {(x,0) : -1
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