1. Let A in R (standard metric) be defined as follows: A={1/n|n € N} U 0 . Is the set B = {b = a +c|a,c € A} Compact?

Q: Exercise. Show that in R with the euclidean metric, any subset of R\Q with more than one element is…

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Q: Show that d given by d(x, y) = |x-ylis a metric on R by using definition 4.2.1. You may want to use…

A: we knew  that given metric d(x,y) = |x-y| is always ≥0.  now d(x,y) = 0 ⇔ |x-y| = 0…

Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric. Hint: As…

A: Answer given below.

Q: Remark. For everyr E N, a set in R" with the Euclidean metric is com- pact if it is bounded and…

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Q: 2. Is the set D = | = {z = C | |z| = 5, |z − i| > 5} compact (C is equipped with a standard metric)?

A: We are given a set defined as . We want to determine if this set is compact in the complex plane…

Q: 2.4.5. Let X be a metric space. Given x E X, prove that {x} is a closed subset of X.

A: Let (X,d) be a metric space to show that {x} is closed sunset of X.

Q: 2. Let (X,r) be a topological space. Where X {a,b.c,d,e} and…

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Q: For the following sets in the metric space (IR, d), determine A' and A. i). (1,2] ii). [0, 1) i).…

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Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric.

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Q: 3- In a complete metric space (X, d), If A C X such that, A = A then A is complete as a sub space of…

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Q: A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

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Q: Let X = R, und d be the discrete metric, und A = (3,5) Show that if A is compact or not

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Q: Let M = 0,2) and d be the usunl metric, then dis not complete O True O False

A: False

Q: Let X = N and d be the usual metric, d (x, y) = |x – yl then d can induces a norm on X True O False…

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Q: Let. f be q into medric Captinyous. marping of a medric rpare %.

A: Matric space

Q: Let p1 and p2 be points in a metric space X. Prove that the set p1 ⋃ p2 is compact in X. Now, let…

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Q: Consider IR as a metric space.with metric dux, y) = 1x-y1. Show that the open interval C0, 1) is an…

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Q: Let X be the space of bounded and closed linear segments on the positive part of the real line, so…

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Q: B. Given a metric space M with metric d, verify that any ε-ball is an open set. (Please If possible…

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Q: A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: Given: H is the set of all points x , y in R2 such that x2+xy +3y2=3. To Show: H is a closed subset…

Q: Q. Let (X, B(R), µ) be a Borel measure space. If AnC = ¢, BnC = ¢, µ(An B) = 0. Determine µ(AUBUC).

A: Measure theory

Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric. Hint: As…

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Q: A. Consider R3 with the metric d((X1, X2, X3), (V1, Y2. Y3) = |X1 - Yıl + |x2 - y2l + [X3 - Y3l. i.…

A: This problem is related with metric on ℝ3. Given that                   dx1,x2,x3,y1,y2,y3 = x1-y1+…

Q: E. Let H be the set of all points (x, y) in R² such that x² + 3y² = 12. Show that H is a closed…

A: We have to find H is closed  subset bounded.

Q: Let d be a metric on X. The distance between an element p E X and a nonempt; subset AC X is defined…

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Q: Q4: Let X = R be a metric space, consider the following subset of X, E = (x €R: 15r<2} 1) Is it a…

A: For the solution follow the next step.

Q: Show that in R with the euclidean metric, any subset of R \Q with more than one element is not…

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Q: Fix a prime number p. For x EQ - {0}, write x = p" () where a, b € Z, a and b are not divisible by…

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Q: QI/ a) Suppose that f: [0, co) y is a metric space. Show that d(x, y) = fCo(x, y)) is a metric on X.…

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Q: Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: As per the question we are given the following bivariate function : x2 + xy + 3y2 = 3 And a set H is…

Q: Consider the sets N = R? \ {(0, y) E R² | |y| > 0}, L = {(x, y) E R² | |x| = |y|}, and let dr,…

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Q: A. Let H be the set of all points (x, y) in R² such that x² + xy + 3y² = 3. Show that H is a closed…

A: To Prove: H=x,y|x2+xy+3y2=3 is a closed set in ℝ2. To Examine: H is bounded or not.

Q: Q. Let (X, B(R), µ) be a Borel measure space. If AnC = 4, BnC = ¢, µ(An B) = 0. Determine µ(A UBUC).…

A: Measure

Q: Exercise. The set N has no accumulation points in R with the euclidean metric.

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Q: "Prove that a function d: A × A → R is precisely a metric on the set A if and only if: 1. For every…

A: Step 1:To prove that a function d:A×A→R is a metric on the set A if and only if it satisfies the…

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric. Hint: As…

A: Given Data:  A set in ℝ2.We need to prove that a given set in ℝ2 is open in the Euclidean metric.

Q: Let X = R, and d be the discrete metric, and A = [3, 6) Show that if A is compact or not

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Q: Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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Q: Let x be an interior point of a subset S of a metric space (X,d). Show that x must also be a limit…

A: Interior Point: Let A be a subset of metric space X, d. A point x∈A is called an interior point of A…

Q: Prove that in any metric space (S, d) every closed ball S,[ro] is a closed set.

A: Use the following definitions, to prove the result. If a set contains all its interior points, then…

Q: In R4, let S = span{(1, 1, 1, 1)T ,(1, 2, 1, 3)T }. Find S⊥.

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