MATH 453
SOLUTIONS TO ASSIGNMENT 5 OCTOBER 8, 2004
Exercise 4 from Section 18, page 111 We check that f is an imbedding; the argument for g is similar. f is clearly a bijection between X and f (X) = X y0 , so we need only show that it and its inverse are
MATH 453
SOLUTIONS TO ASSIGNMENT 8 NOVEMBER 6, 2004
Exercise 2 from Section 26, page 171 (a) Let X be a subspace of R in the finite complement topology, and let cfw_U be an open cover for X. Pick a particular U0 = R - A0 , where A0 is finite. Then X - U0
MATH 453
SOLUTIONS TO ASSIGNMENT 3 SEPTEMBER 26, 2004
Exercise 1 from Sections 14-16, page 91 First consider A as a subspace of Y. In this topology, a subset B of A is open if and only if it is of the form B = A V, where V is an open set in Y. Now Y is a
MATH 453
SOLUTIONS TO ASSIGNMENT 2 SEPTEMBER 20, 2004
Exercise 1 from Sections 12 & 13, page 83 We will show that A is open by exhibiting it as a union of open sets. For each x A, let U x be the open set containing x such that U x A. It is easy to see tha
MATH 453
SOLUTIONS TO ASSIGNMENT 6 OCTOBER 16, 2004
Exercise 4 from Section 20, page 127 (a) Since each of their component functions are continuous, all three of f , g and h are continuous when R is given the product topology. On the other hand, if R is g
MATH 453
SOLUTIONS TO ASSIGNMENT 10 NOVEMBER 23, 2004
Exercise 5 from Section 30, page 194 (a) Let D be a countable dense subset of the metrizable space X. I claim that B = cfw_B(x, 1/n) | x D and n Z+ is a countable basis. First, B is countable, since b
MATH 453
SOLUTIONS TO ASSIGNMENT 11 DECEMBER 3, 2004
Exercise 2 from Section 51, page 330 This is a special case of problem 3 below. Exercise 3 from Section 51, page 330 (a) The formula F(x, t) = (1 - t)x is a homotopy between the identity map on either I
MATH 453
SOLUTIONS TO ASSIGNMENT 1 SEPTEMBER 8, 2004
Exercise 4 from Section 3, page 28 (a) Reflexivity, symmetry and transitivity for follow from the corresponding properties for equality, so is an equivalence relation. For instance, to check symmetry, w
MATH 453
SOLUTIONS TO ASSIGNMENT 9 NOVEMBER 11, 2004
Exercise 6 from Section 28, page 181 X is a metric space, so it is Hausdorff. Thus X is compact Hausdorff and we need only check that f is a continuous bijection. Continuity and injectivity is immediate
MATH 453
SOLUTIONS TO ASSIGNMENT 4 OCTOBER 7, 2004
Exercise 2 from Section 17, page 100 Using Theorem 17.2, A is closed in Y implies that it is the intersection of Y and a closed subset C of X: A = C Y. Since Y is given to be closed in X, A is the interse
Part IV: You find that a small business loan in the amount of 50,000 is the amount you need to purchase the restaurant location. After researching banks to find the best interest rate, you find that banks for small businesses offer the best interest rate
MATH 453
SOLUTIONS TO ASSIGNMENT 7 NOVEMBER 1, 2004
Exercise 3 from Section 23, page 152 Let B = A A . Then B is connected for each since A and A are connect and have a point in common. Now B = A A and the B 's have a point in common (A is nonempty) so A
Math W4051: Problem Set 1 due Wednesday, September 12 1. Let d1 and d2 be two metrics on the same set X with the property that there exist constants C, C > 0 such that C d1 (x, y) d2 (x, y) C d1 (x, y), for all x, y X. (a) Show that a sequence cfw_xn of
Math W4051: Problem Set 11 due Wednesday, December 5 Reading: Munkres, Ch.13. Also take a look at Theorem 54.6 in Ch.9. 1. Express the following abelian groups in standard form: (a) Ab x, y | x2 = y 3 ; (b) Ab x, y | x2 = y 4 . 2. Let X and Y be manifolds
Math W4051: Problem Set 10 due Wednesday, November 28 Reading: Munkres, Ch.9 59-60, Ch.11. 1. Show that x, y | xyxy = 1 is isomorphic to Z (Z/2Z). 2. Show that x, y, z | xyz = zyx is isomorphic to Z (Z Z). 3. Show that x, y | x2 = y 3 is isomorphic to x,
Math W4051: Problem Set 9 due Wednesday, November 21 1. Munkres, Ch.9 58, exercise 2 on p. 366. For each of the spaces in (a), (c), (d), (f), (g), (h), (i), you need to write down explicitly the formula for a deformation retraction (or some other form of
Math W4051: Problem Set 8 due Wednesday, November 14 Reading: Munkres, Ch.9 55-58. 1. Munkres, Ch.9 55, exercises 2, 4(a), 4(b), 4(c) and 4(d) on p. 353. 2. Let f, g : [0, 1] [0, 1]2 be two continuous paths in a square such that f (0) = (0, 0), f (1) = (1
Math W4051: Problem Set 7 due Wednesday, November 7 Reading: Munkres, Ch.9 51-54. 1. Munkres, Ch.9 51, exercises 3(a) and 3(b) on p. 330. 2. Munkres, Ch.9 52, exercise 3 on p. 335. 3. Munkres, Ch.9 53, exercises 3 and 5 on p. 341. 4. Munkres, Ch.9 54, exe
Math W4051: Problem Set 6 due Wednesday, October 17 Reading: Munkres, Ch.4 36, Ch.12 76-78. 1. Munkres, Ch. 4 32, exercise 3 on p. 205. 2. Munkres, Ch.12 78, exercises 2(b), 2(c),3,4 on p. 476. You can assume Part 2(a) when solving 2(b) and 2(c). 3. Using
Math W4051: Problem Set 5 due Wednesday, October 10 Reading: Munkres, Ch.3 26-29, Ch.4 30-32, Ch.7 43-44. 1. Munkres, Ch.3 26, exercise 2 on p. 171. 2. Munkres, Ch.3 29, exercise 1 on p. 186. 3. Munkres, Ch.7 43, exercise 5 on p. 270. 4. Let X be the real
Math W4051: Problem Set 4 due Wednesday, October 3 Reading: Munkres, Ch.3. 1. Define an equivalence relation on the space X = R2 - cfw_0 as follows: (x1 , y1 ) (x2 , y2 ) if and only if there exists k Z such that (x1 , y1 ) = (2k x2 , 2k y2 ). The space X
Math W4051: Problem Set 3 due Wednesday, September 26 Reading: Munkres, Ch.2, 18-22. 1. Munkres, Ch.2 17, exercises 6, 7, 8, 14, 15, 20 on p. 100-102. 2. Munkres, Ch.2 19, exercises 3, 7 on p. 118. 3. Munkres, Ch.2 20, exercise 5 on p. 127. 4. Recall the
Math W4051: Problem Set 2 due Wednesday, September 19 Reading: Munkres, Ch.1, 7, and Ch.2, 12-17. 1. Munkres 13, exercises 1 and 3 on p.83. 2. Munkres 16, exercise 3 on p. 92. Note: All the spaces below are endowed with the restriction of the Euclidean me
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