(b) (i) Give an example of an infinite collection of nested open sets in R under the usual metric on R such that their intersection is closed and non-empty.

Q: Give an example of an infinite collection of nested open sets in RR under the usual metric on RR…

A: We given an example of infinite collection of nested open sets in R such that their intersection is…

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

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

A:

Q: Q7: Prove that (R", |l-|| ) is complete metric space.

A: This is a problem of Topology, Functional Analysis.

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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Q: Let X be a topological space having an open cover U1,..., Un with the property that all k-fold…

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Q: show that any infinite discrete topological space is not compact

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Q: a) Show that every closed subset of a compact space is compact but the converse need not to be true…

A: As per the guidelines I had given the proof of the first theorem. Kindly repost the another theorem…

Q: The relation R defined on a set A is antisymmetric if (a, b) ER = (b, a) E R for (A) Every (a, b) ER…

A: A relation R is said to be antisymmetric if (a,b)∈R and (b,a)∈R implies a=b

Q: 2.4.10. Let X be a metric space. (a) Suppose that X is complete and F1 2 F2 2 ··. is a nested…

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Q: (3) Let R² and R be endowed with their Euclidean metrics. (a) Show that projections p₁(x, y) = x and…

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Q: (a) Let [a, b], a <b, be any closed interval. Prove that |[a, b]| = |[0, 1].

A: Let  be any closed interval.The main aim is to prove that :"As per the policy we are allowed to…

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Q: Is not closed set in X x X. Then the space X is: (a) Hausdorff Space (b) Not Hausdorff Space (c)…

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Q: Read the graph pooot of the closed theorem carefully ounel explain wher nd how we have used facts in…

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Q: 1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B → R by…

A: we have to solve given problem:

Q: 2) Give an example of a metric space such that the closed ball Bå(x,r) is not an open set for any x…

A: To Find: Give an example of a metric space such that a closed ball B¯d(x,r) is not open set for any…

Q: Theorem 2.1. Let{U}{_, be a finite collection of open sets in a topological space (X,T). Then N, U…

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Q: (i) Show that a subset A of a topological space X is closed if and only A = A (ii) Prove that A…

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Q: EXERCISE 2: For each a/ and b/ answer these questions: (i) Sketch the set of points (directly, no…

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Q: Suppose X is a connected topological space with the property that every point x of X has a…

A: In this problem, we consider a connected topological space X with the property that every point in X…

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Q: Theorem 2.25. Suppose that (X, p) is a metric space. (a) The intersection of any collection of…

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A: A can be written in interval form

Q: Provide an example of a topological space that is connected but not path-connected,why?

A: We use definition of connected and path connected sets. We use contradiction method.

Q: Let T be a topology on X. Assume that T is Hausdorff and let x € X. (1) Show that {x} (and hence…

A: let T be a topology on x. T is a Hausdorff space

Q: Exercise 1. Let X be a metric space. The subsets of X which are arbitrary (finite or infinite)…

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Q: Suppose A is a closed, non-empty subset of a metric space (X, d) and p a point of X which is not in…

A: Given:  A is a closed and non-empty set of a metric space (X,d) and p∈X , p∉A. To Prove: inf{d(p,a)…

Q: .. .): P, is a polynomial space with degree with order n and less than equal n. Is the space S =…

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Q: Consider relation R={(x,y)|x divides y,xEA,y€A} on the set A={2,3,4,5,10,12,25,30}. (a) Check…

A: We are given a relation : R={(x,y)|x divides y,x∈A,y∈A}  on a set A={2,3,4,5,10,12,25,30}

Q: (a) Prove that every closed subset S of a compact metric space (M,d) is compact in M.

A: Note: Our guidelines we are supposed to answer only one question. Kindly repost other question as…

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Q: Let X be a set with more than one element. (X,7) is a topological space such that Vxe X, {x} €T then…

A: Given the set X contains more than one element.