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: Hwi Let A () Find all eigonvotveo of A Gnd their corispondhing eigevenvedters

A: Topic:- matrix

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: (iv) If a sequence {n} in X converges, it converges to a unique point. (iv) Every convergent…

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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: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = A n (A)° = A int A (ii)…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 11. If a set has empty interior, then so does its closure. 12. The closure of a set coincides with…

A: We need to show whether the given statement "If a set has empty interior, then so does it's closure"…

Q: In each part of this question, you are asked to give an example of something, and explain why it is…

A: (a) Consider here      Noted that here  are limit points of  .Because here if write the some…

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: 3.1 Find all limit points of a set A = {(2+ ,2 -n E N} on the Euclidean plane with the usual metric.…

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Q: Give an example of a set w-limit that is a minimal set.

A: To give an example of a set w-limit that is a minimal set.  Given a collection of sets , a member…

Q: Let R be equipped with the Euclidean topology T and let Y =]10,20[. We denote by Ty the induced…

A: According to our guidelines we can answer only three subparts, or first question and rest can be…

Q: If a < b, then [a,b] CR does not have con- tent 0. In fact, if U1,. . .,Un) is a finite cover of…

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

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

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Q: 2. Let X be a normed space. Show that X is a Banach space if and only if the unit sphere Sx = {x € X…

A: To prove the given theorem.

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

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Q: Let two metrics be bi-lipschitz, prove it is an equivalence relation

A: Find the attachment for the solution.

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: (b) A function f: (0, 1) → R that is continuous at all a € (0, 1) except at the points an = for n€…

A: (a) Consider here      Noted that here  are limit points of  .Because here if write the some…

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. A non-empty complete and bounded metric space (X, d) with exactly a countably infinite (not…

A: To Find: An example of a complete and bounded metric space (X,d) with countably infinite number of…

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: Let S be a bounded nonempty subset of R and suppose that inf S /∈ S. Prove that there is a…

A: Step 1: Let s=supS.  So,  for any ε&gt;0, there exists an x∈S such that s−ε&lt;x.  Therefore, for…

Q: (ii) Check whether or not each of the following interval is a neighborhood of 0 under usual topology…

A: (ii).  (iii).

Q: Which one of the following statements is true?

A: In the R with the Euclidean topology and R with the finite closed topology, f might not be…

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: 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: (2.1) Let b> 0 and X [-b, b). Here we admit %3D B"open" in r if and only if (0} B or (-b, b) C B.…

A:

Q: Answer selection  1) 2 2) the limit does not exist  3) 1.5 4) 1

A:

Q: To identify how much heat is required to bring a kettle of water to its boiling point, you are…

A: Here,

Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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

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Q: 6. Let (X, d) be a metric space and Y CX be nonempty. Verify that (Y,d) is a metric space. Show that…

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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: please answer fully on paper and make sure its correct please i keep getting different answers

A: