1. Suppose that {an} and {b.) are bounded sequences. Show that lim sup an lim supbn. lim sup an +bn 2. Show that R with the Euclidean metric is the union of countably many bounded sets. Is that true in any metric space?

Q: OConsider the subset A = {1,, ., , ..} of R, the set of all real numbers 1 1 with the euclidean…

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Q: (g) Give an example of an infinite collection of nested open sets in R under the usual metric on R…

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

A: 3.3 is also known as Hausdorff Property. d(x1, x2) > 0 for every x1 ≠ x2   (metric space…

Q: 3. Let (X, d) be a metric space and x1, x2, ... , Xn, ... a sequenc

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Q: Consider the space Z+ with the finite complement topology. Consider the sequence (xn) of points in…

A: (ℤ , +) with finite complement topology. (ℤ+ , τC)   where τC={U⊆ℤ+ |ℤ+ ∖U is finite } The sequence…

Q: 1. Prove each of the following for a metric space (X, d): (i) A set A C X is closed if and only if…

A: We need to prove the following for the metric space (X,d): (i) For A⊆X is closed if and only if for…

Q: Give an example of a metric space X, d and a sequence {Pn} C X such that {pn} is a Cauchy sequence…

A: Let X=(0,∞) Define d:X×X→ℝ+∪0 by d(x,y)=x-y.i.e. d is a standard metric on X induced from ℝ.Take…

Q: 3- Give an example to show that a converg ent sequence under one metric may be divergent und er a…

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Q: If there exists open ball B (a, r) such that B (a,r) nG = Ø then a is not limit point of G True…

A: TRUE

Q: Let (X, d) be a metric space. Suppose that (xn) is a sequence of points in X which converges to a…

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Q: Let (Xd) be a metric space and suppose that a sequence (a.} has a subsequence that converges to an…

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Q: is Q:- show that the space x = [0₂e) complete space, where e= exponential nat je c= 2.71898

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Q: Explain the bolzano –Weierstrass theorem. Every infinite and bounded set has atleast one limit point

A: Bolzano-Weierstrass theorem:-  Every infinite and bounded set has atleast one limit point

Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

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Q: (A,7) be the topological space defined in question 2. Let A = {1,2,3}, then the set of limit points…

A: Solution: (4) Answer: The set is  A=1,2,3 Clearly, set A is finite.  We know that a finite set does…

Q: 5. Which of the following conditions is sufficient to conclude that a subset F of a metric space is…

A: We want to find sufficient condition to conclude that F is closed.

Q: Please use implicit differentiation so that I can understand the concept better.

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Q: 5. Let (X, d) be a metric space such that the intersection of open set and closed set is always…

A: As per the question we are given a metric space (X, d) space such that the intersection of open set…

Q: 1+n Prove using the definition of the limit that limn→∞ 1-2n 2

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Q: 2. Let X be a non-empty set. Define the discrete metric on X and answer each of the following: (i)…

A: First we have to define the discrete metric on X. Definition (Discrete Metric): The discrete metric…

Q: 4.1 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set. 4.2 Let F be a…

A: Given (S,d) is a metric space. Three questions related to this are given. The solution to these…

Q: Question 2. space is the union of a finite number of closed balls radius e. Prove that a metric…

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Q: Ok so the question says " evaluate. Write your answer as a fraction or whole number without…

A: The negative exponent rule: Tells us that the number with a negative exponent should be put to the…

Q: 1. Show that any interval (a,b) in R with the discrete metric is locaaly compact but not compact

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Prove

A: :

Q: (2) If SC R" prove that the collection of isolated points of S is count- able (hint: use a covering…

A: To show  that  collection of isolated points of a  subset    of    is countable. We  can  use Open…

Q: Given a metric space M with metric d, verify that any e-ball is an open set.

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Q: thun no in

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Q: Corollary 2.53 Let (M, d) be a metric space and K C M. The following statements are equivalent: (i)…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove that ℝ with lower limit topology is Hausdorff

A: Solution :-

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: 3. a. Define the following with an example: i. paths ii. simple graph 0- 3. 2- b.Draw the graph with…

A: As per our guidelines I am solving only 1st for others kindly resubmit the question mentioning that…

Q: 2. State and prove the Cantor's Intersection Theorem.

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Q: 3. Let a, b e R such that a < b. Prove that the open interval (a, b) is an open set in R.

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Q: Suppose poset (A, R), where A = {1, 2, 3, 4, 6, 8. 9. 12, 16, 18, 32, 36} and the relation R runs…

A: With the help of definition we solve the given problem as follows:

Q: 5. Let (X, d) be a compact metric space and {K) a decreasing sequence of closed sets in X. If : XX…

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Q: d((X1, X2, X3), (V₁, V2, Y3) = |X₁ - Y₁| + |×2 - Y₂ + X3 - Y3l. Conclude that, with this metric, a…

A: Given  is a metric space.To show that a subset  of  is sequentially compact if and only if it is…