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In this problem we show that any metric space (X, d) is homeomorphic to a bounded metric space. (a) Define ρ : X X R by...
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...
Please prove Theorem 7.10: Show for any open intervals (a, b) and (c, d) in R that ((a, b), U(a, b) and ((c, d), Uc, d)) are homeomorphic. (Hint: Find a linear function f: (a, b)- (c, d) for which f(...
Please prove
Theorem 7.10: Show for any open intervals (a, b) and (c, d) in R that ((a, b), U(a, b) and ((c, d), Uc, d)) are homeomorphic. (Hint: Find a linear function f: (a, b)- (c, d) for which f(a)-c and f(b)-d and show this is a homeomorphism.)
Theorem 7.10: Show for any open intervals (a, b) and (c, d) in R that ((a, b), U(a, b) and ((c, d), Uc, d)) are homeomorphic. (Hint: Find a linear function...
Topology 3. Either prove or disprove each of the following statements: (a) If d and p...
Topology
3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
7. Recall the space m of bounded sequences of real numbers together with the metric d(х,...
7. Recall the space m of bounded sequences of real numbers together with the metric d(х, у) — suр |2; — Ук). k 1,2. (a) Give a simple proof to show that m is complete by showing that m = suitable space X. (Recall that C(X) denotes the space of continuous bounded real- valued functions on X together with the supremum norm.) C(X) for some (b) Let A denote the unit ball in m given by А 3 (x€ т:...
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....
A metric space (X, d) is totally bounded if, given ε>0, there exists a finite subset...
A metric space (X, d) is totally bounded if, given
ε>0, there exists a finite subset =
of X, called an ε-net, such that for each x∈X there
exists
∈
such that d(x,)
< ε. Prove that if Y is a subset of a totally bounded space X
then, given ε>0, the subset Y has an ε-net and
therefore Y is also totally bounded.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...
Metric Space Enrichment Problem E2.1. In this problem, we consider a new metric not considered in...
Metric Space
Enrichment Problem E2.1. In this problem, we consider a new metric not considered in the lectures. We set X = Q the set of rational numbers, and fix a prime number p E N. For 91.92 € Q, we define 8(91,92) (the p-adic distance) as follows. If qı = 42, we define 8p(91,92) = 0. Otherwise, write 91 - 92 = a/b+ 0 in lowest terms. At most one of a and b can contain a power of...
We define the set X ⊆ L^∞ by X = { {xn} ∈ L^∞ : lim n→∞ xn = 1 } Prove that the set X with the subspace metric d∞|X×X is a complete metric space.
We define the set X ⊆ L^∞ by X = { {xn} ∈ L^∞ : lim n→∞ xn = 1 } Prove that the set X with the subspace metric d∞|X×X is a complete metric space.
Please answer c d e 3. This problem shows that the metric space of continuous real-valued...
Please answer c d e
3. This problem shows that the metric space of continuous real-valued functions C([0, 1]) on the interval [0, 1is complete. Recall that we use the sup metric on C([0,1), so that df, 9) = sup{f (2) - 9(2): € (0,1]} (a) Suppose that {n} is a Cauchy sequence in C([0,1]). Show that for each a in 0,1], {Sn(a)} is a Cauchy sequence of real numbers. (b) Show that the sequence {fn(a)} converges. We define f(a)...
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.
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...
Please prove
Theorem 7.10: Show for any open intervals (a, b) and (c, d) in R that ((a, b), U(a, b) and ((c, d), Uc, d)) are homeomorphic. (Hint: Find a linear function f: (a, b)- (c, d) for which f(a)-c and f(b)-d and show this is a homeomorphism.)
Theorem 7.10: Show for any open intervals (a, b) and (c, d) in R that ((a, b), U(a, b) and ((c, d), Uc, d)) are homeomorphic. (Hint: Find a linear function...
Topology
3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
7. Recall the space m of bounded sequences of real numbers together with the metric d(х, у) — suр |2; — Ук). k 1,2. (a) Give a simple proof to show that m is complete by showing that m = suitable space X. (Recall that C(X) denotes the space of continuous bounded real- valued functions on X together with the supremum norm.) C(X) for some (b) Let A denote the unit ball in m given by А 3 (x€ т:...
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....
A metric space (X, d) is totally bounded if, given
ε>0, there exists a finite subset =
of X, called an ε-net, such that for each x∈X there
exists
∈
such that d(x,)
< ε. Prove that if Y is a subset of a totally bounded space X
then, given ε>0, the subset Y has an ε-net and
therefore Y is also totally bounded.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...
Metric Space
Enrichment Problem E2.1. In this problem, we consider a new metric not considered in the lectures. We set X = Q the set of rational numbers, and fix a prime number p E N. For 91.92 € Q, we define 8(91,92) (the p-adic distance) as follows. If qı = 42, we define 8p(91,92) = 0. Otherwise, write 91 - 92 = a/b+ 0 in lowest terms. At most one of a and b can contain a power of...
Please answer c d e
3. This problem shows that the metric space of continuous real-valued functions C([0, 1]) on the interval [0, 1is complete. Recall that we use the sup metric on C([0,1), so that df, 9) = sup{f (2) - 9(2): € (0,1]} (a) Suppose that {n} is a Cauchy sequence in C([0,1]). Show that for each a in 0,1], {Sn(a)} is a Cauchy sequence of real numbers. (b) Show that the sequence {fn(a)} converges. We define f(a)...
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.
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