Homework Answers
Problem 1. Let (X, d) be a metric space and t the metric topology on X....
Problem 1. Let (X, d) be a metric space and t the metric topology on X. (a) Fix a E X. Prove that the map f :(X, T) + R defined by f(x) = d(a, x) is continuous. (b) If {x'n} and {yn} are Cauchy sequences, prove that {d(In, Yn)} is a Cauchy sequence in R.
#9. all one problem. then e is 0U tric space. Show that there is an isometric...
#9. all one problem.
then e is 0U tric space. Show that there is an isometric imbedding h of X 、D), as follows: Let X denote the set of all space () into a complete metric Cauchy sequences of points of X. Define x~ y if Let [x] denote the equivalence class of x; and let Y denote the set of these eq x (xi, x2, ...) uiv alence classes. Define a metric D on Y by the equation linnod(xn,...
5. Let X be a metric space. (a) Let x E X be an isolated point....
5. Let X be a metric space. (a) Let x E X be an isolated point. Prove that the only sequences in X that converge to a are the sequences that are eventually constant (b) Prove that the only convergent sequences in a discrete metric space (See Problem 8 on page 79 for the definitions of "isolated" and "discrete.") with tail a,z,x.... are the eventually constant sequences.
A subset D of a metric space (X, d) is dense if every member of X...
A subset D of a metric space (X, d) is dense if every member of
X is a limit of a sequence of elements from D.
Suppose (X,d) and (Y,ρ) are metric spaces and D is a dense
subset of X.
1. Prove that if f : D -» Y is uniformly continuous then there exists an extension15 of f to a if dn (E D) e X define 7(x) lim f(d,) uniformly continuous function f:X * Y. Hint: 2....
Question 1. Give an example of a complete metric space (X, d) and a function f...
Question 1. Give an example of a complete metric space (X, d) and a function f :X + X such that d(f(x), f(y)) < d(x, y) for all x, y E X with x + y and yet f does not have a fixed point. a map f:X + X has a fixed point if there is an element a E X such that f(a) = a.
A. Let (X, d) be a metric space so that for every E X and every...
A. Let (X, d) be a metric space so that for every E X and every r>0 the closed ball N,(z) = {ye X : d(y, z) < r} is com pact. Let be a homeomorphism. (1) Prove that f"-+m-fn。fm for all n, m E Z. (2) Let z E X and suppose that F, {fn (z) : n E 2) is a closed subset of X Prove that F is a discrete subset of X (A subset Y C...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
In the following, (X,d) is an arbitrary metric space and (X,d,μ) is an arbitrary metric measure...
In the following, (X,d) is an arbitrary metric space and (X,d,μ)
is an arbitrary metric measure space.
(6) Recall the definition of bounded set: The set A C (X, d) is bounded if δ(A) < 00 where 6(A)p d(a,a). (X,d) with ACBand B is bounded then A is bounded (a) Show that if A, B (b) Fix a set A. I B - (r), a single point, show that D(A, B)-0 if and only f (c) Prove that the function...
8. [10 pts] (Carothers 7.46.) If A is a dense subset of a metric space (M, d), show that (A, d) a...
8. [10 pts] (Carothers 7.46.) If A is a dense subset of a metric space (M, d), show that (A, d) and (M, d) have the same completion (isometrically). Hint: If M is the completion for M, then A is dense in M. Why?] Note. As in our proof of Theorem 7.18, and as Carothers does in the hint, you can take M to be a subset of M (i.e., not just isometric to one).
8. [10 pts] (Carothers 7.46.)...
Q3 * You are given two subsets A, B C M of a metric space M....
Q3 * You are given two subsets A, B C M of a metric space M. Define p by p=inf{d(x, y)| 2 € A, Y E B}. Prove that, if p > 0 then A and B are separated. Give an example where p=0, but A and B are not separated. Q4 Show that Q as a subset of R is disconnected. Likewise show R Q is disconnected.
Problem 1. Let (X, d) be a metric space and t the metric topology on X. (a) Fix a E X. Prove that the map f :(X, T) + R defined by f(x) = d(a, x) is continuous. (b) If {x'n} and {yn} are Cauchy sequences, prove that {d(In, Yn)} is a Cauchy sequence in R.
#9. all one problem.
then e is 0U tric space. Show that there is an isometric imbedding h of X 、D), as follows: Let X denote the set of all space () into a complete metric Cauchy sequences of points of X. Define x~ y if Let [x] denote the equivalence class of x; and let Y denote the set of these eq x (xi, x2, ...) uiv alence classes. Define a metric D on Y by the equation linnod(xn,...
5. Let X be a metric space. (a) Let x E X be an isolated point. Prove that the only sequences in X that converge to a are the sequences that are eventually constant (b) Prove that the only convergent sequences in a discrete metric space (See Problem 8 on page 79 for the definitions of "isolated" and "discrete.") with tail a,z,x.... are the eventually constant sequences.
A subset D of a metric space (X, d) is dense if every member of
X is a limit of a sequence of elements from D.
Suppose (X,d) and (Y,ρ) are metric spaces and D is a dense
subset of X.
1. Prove that if f : D -» Y is uniformly continuous then there exists an extension15 of f to a if dn (E D) e X define 7(x) lim f(d,) uniformly continuous function f:X * Y. Hint: 2....
Question 1. Give an example of a complete metric space (X, d) and a function f :X + X such that d(f(x), f(y)) < d(x, y) for all x, y E X with x + y and yet f does not have a fixed point. a map f:X + X has a fixed point if there is an element a E X such that f(a) = a.
A. Let (X, d) be a metric space so that for every E X and every r>0 the closed ball N,(z) = {ye X : d(y, z) < r} is com pact. Let be a homeomorphism. (1) Prove that f"-+m-fn。fm for all n, m E Z. (2) Let z E X and suppose that F, {fn (z) : n E 2) is a closed subset of X Prove that F is a discrete subset of X (A subset Y C...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
In the following, (X,d) is an arbitrary metric space and (X,d,μ)
is an arbitrary metric measure space.
(6) Recall the definition of bounded set: The set A C (X, d) is bounded if δ(A) < 00 where 6(A)p d(a,a). (X,d) with ACBand B is bounded then A is bounded (a) Show that if A, B (b) Fix a set A. I B - (r), a single point, show that D(A, B)-0 if and only f (c) Prove that the function...
8. [10 pts] (Carothers 7.46.) If A is a dense subset of a metric space (M, d), show that (A, d) and (M, d) have the same completion (isometrically). Hint: If M is the completion for M, then A is dense in M. Why?] Note. As in our proof of Theorem 7.18, and as Carothers does in the hint, you can take M to be a subset of M (i.e., not just isometric to one).
8. [10 pts] (Carothers 7.46.)...
Q3 * You are given two subsets A, B C M of a metric space M. Define p by p=inf{d(x, y)| 2 € A, Y E B}. Prove that, if p > 0 then A and B are separated. Give an example where p=0, but A and B are not separated. Q4 Show that Q as a subset of R is disconnected. Likewise show R Q is disconnected.
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