Let (X, d) be a discrete space and let (Y, d′) be any metric space. Prove...
Let (X, d) be a discrete space and let (Y, d′) be any metric space.
Prove that any function f : (X, d) → (Y, d′) is continuous.
(Namely, any function from a discrete space to any metric space is continuous.)
Homework Answers
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Let (X, d) be a discrete space and let (Y, d′) be any metric
space.
Prove...
Let (X, d) be an infinite discrete metric space. Prove that any infinite subset of X...
Let (X, d) be an infinite discrete metric space. Prove that any infinite subset of X is closed and bounded but NOT compact
Let (X, d) be a compact metric space that has at least two elements. Prove that...
Let (X, d) be a compact metric space that has at least two elements. Prove that if y ∈ X then fy(x) = d(x, y) is a continuous function fy : X → R.
(TOPOLOGY) Prove the following using the defintion: Exercise 56. Let (M, d) be a metric space...
(TOPOLOGY) Prove the following using the defintion:
Exercise 56. Let (M, d) be a metric space and let k be a positive real number. We have shown that the function dk defined by dx(x, y) = kd(x,y) is a metric on M. Let Me denote M with metric d and let M denote M with metric dk. 1. Let f: Md+Mk be defined by f(x) = r. Show that f is continuous. 2. Let g: Mx + Md be defined...
(a) Let (X, d) be a metric space. Prove that the complement of any finite set...
(a) Let (X, d) be a metric space. Prove that the complement of any finite set F C X is open. Note: The empty set is open. (b) Let X be a set containing infinitely many elements, and let d be a metric on X. Prove that X contains an open set U such that U and its complement UC = X\U are both infinite.
Jet f be continuons one to one m compact metric space X onto a metric space...
Jet f be continuons one to one m compact metric space X onto a metric space Y. Prove that f'Y ~ X is continuoms (Hint: use this let X and Y e metric space, and let f be function from X to Y which is one to one and onto then the following three statments are equivalent. frs open, f is closed, f is continuous.
1. (a) Let d be a metric on a non-empty set X. Prove that each of...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
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.
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....
9. Let X and Y be metric spaces, and let D be a dense subset of...
9. Let X and Y be metric spaces, and let D be a dense subset of X. (For the definition of "dense, see Problem 4 at the end of Section 3.5.) (a) Let f : X → Y and g : X → Y be continuous functions. Suppose that f(d)gld) for all d E D. Prove that f and g are the same function.
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) =...
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) = / If(x)-g(x)| dx. 9) Let (X, di) and (Y, d2) be metric spaces. a) Prove that X × Y is a metric space with the metric b) Prove that X x Y is a metric space with the metric
(TOPOLOGY) Prove the following using the defintion:
Exercise 56. Let (M, d) be a metric space and let k be a positive real number. We have shown that the function dk defined by dx(x, y) = kd(x,y) is a metric on M. Let Me denote M with metric d and let M denote M with metric dk. 1. Let f: Md+Mk be defined by f(x) = r. Show that f is continuous. 2. Let g: Mx + Md be defined...
(a) Let (X, d) be a metric space. Prove that the complement of any finite set F C X is open. Note: The empty set is open. (b) Let X be a set containing infinitely many elements, and let d be a metric on X. Prove that X contains an open set U such that U and its complement UC = X\U are both infinite.
Jet f be continuons one to one m compact metric space X onto a metric space Y. Prove that f'Y ~ X is continuoms (Hint: use this let X and Y e metric space, and let f be function from X to Y which is one to one and onto then the following three statments are equivalent. frs open, f is closed, f is continuous.
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
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.
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....
9. Let X and Y be metric spaces, and let D be a dense subset of X. (For the definition of "dense, see Problem 4 at the end of Section 3.5.) (a) Let f : X → Y and g : X → Y be continuous functions. Suppose that f(d)gld) for all d E D. Prove that f and g are the same function.
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) = / If(x)-g(x)| dx. 9) Let (X, di) and (Y, d2) be metric spaces. a) Prove that X × Y is a metric space with the metric b) Prove that X x Y is a metric space with the metric
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