If A is a compact subset of a metric space (X, d) and B is a closed subset of A, prove that B is also compact.

Q: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

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

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

Q: Let (X, d) be a metric space and let A ⊆ X be a compact subset. show that, If B ⊆ X and d(A, B) =…

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Q: Let (X, d) be a metric space with X being an infinite countable set. Show that X is not connected.

A: A set is said to be infinite countable if its elements can be counted like natural number but you…

Q: The only subsets of R which are both open and closed under the discrete metric are Ø and R. Select…

A: This is False statement, in discreate metric space, every subset of R is both open as well as…

Q: Let E1, E2, ..., En ⊂ X,d be compact subsets of a metric space X. Just using the definition of…

A: For a metric space we know that the for a compact set every open cover has a finite subcover. So…

Q: Let (X, d) be a metric space with X being a countable set. Show that X is not connected.

A: Let (X, d) be a metric space with X being a countable set. Show that X is not connected.

Q: Let (X, I) be a topological space and suppose that A and B are subsets of X such that ACB. Prove…

A: Let x ∈A°. Then by definition, there exists an open set U  such that  x∈U⊆A.Since A⊆B hence x∈U⊆B ,…

Q: For a normed space X and a closed convex subset A of X. Prove that (º )nÎ= Â. A

A: Let, be a normed space and be the closed convex subset of As, we know that norm on is a function…

Q: Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

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Q: (a) Prove that every finite subset of a metric space is closed and has umulation points.

A: Since you have posted multiple questions, but according to guidelines we will solve the first…

Q: If S is a closed bounded subset of a metric space X, then S is compact.

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Q: Let (X,d) be any metric space, then the only open sets and closed sets are and X O True O False

A: Given statement is false because if we take descrete metric space

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If ACB,…

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Q: The set [0, 1] with the discrete metric is compact. O True O False

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Q: Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only…

A: Given: X,d is a metric space, x∈X and A⊂ X is a non-empty set. To Prove: dx,A=0 if and only if every…

Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

Q: If M and N are metric spaces, the Cartesian product M x N is again a netric space with the distance…

A: We have given that , M and N are metric spaces. We have to show that , M × N is again a metric space…

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

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Q: let C be a compact subset in a Hausdorff topological space (X,T) and b∈X a point. If b∉C then prove…

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Q: Show that any infinite subset A of a discrete topological space X is not compact.

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Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

Q: Is the following statement True or False? Justify each answer. If S is a closed bounded subset of a…

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Q: If x is a limit point of a subset A of a metric space X, then sh

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Q: Let E and F be compact sets in a metric space (X, d). Show that EUF is compact using the definition…

A: 1. Let E and F be compact subsets in a metric space X,d. Let UE and UF be arbitrary open covers of E…

Q: Let (C[a,b] .d) be the discrete metric space. Then Cla,b] is bounded. * False True

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Q: If A is a subset of a metric space X, then show that A = (x: each neighbourhood of x intersects A}

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Q: Let K be a compact subset of a metric space X, and assume that -R is continuous. Prove that f…

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Q: Q7: Prove that ( IR", ||·|| ) is complete metric space.

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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…

Q: 1. Let M, d be a metric space with A C B C M. Suppose that B is totally bounded. Prove that A is…

A: Let (M, d) be a metric space and let A be a nonempty subset of M. The diameter of A is defined as…

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

A: This is a problem of Functional Analysis.

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

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Q: Prove by sequences that the open balls in R^n with the usual (Euclidean) metric are not compact

A: In this question, we have to show that open balls in ℝn with the usual metric are not compact by the…

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: Let M, be set of all rare subsets of a metric space X. Prove that countable union of M, is rare.

A: As per bartleby guidelines for more than one question asked only first should be answered. Please…

Q: (d) Every continuous map on any metric space is an open map.

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Q: Every closed subset of a connected metric space is connected. O True O False

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Q: Let (R, d) be discrete metric space and A = N then A is complete and compact neither complete nor…

A: Given that (ℝ,d) is a discrete matric space and A=ℕ. then     d(x,y)=0 if x=y1  if x≠y A matric…

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

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Q: Amongst the various possible products of four topological spaces A, B, C and D, show that (A × (B ×…

A: We shall solve this in next step

Q: Prove that if X,d is a metric space and G ⊆ H ⊆ X, then G-bar ⊆ H-bar.

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Q: H.W.2 Exercise Prove that: 1. (IR, J- |) is a complete metric space. 2. (Q. |-1) is not complete…

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Q: Q6: (a) In any metric space (S, d), prove that every compact subset T is complete. (b) Prove that…

A: A metric space is a set together with a metric on the set. The metric is a function that defines a…

Q: Let (X, d) be a metric space and let A be a non-empty subset of X. Prove that A is open if and only…

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

A: Given that M is a metric space with metric d. Let Ba,ε be an ε-ball with center at a. That is:…

Q: the euclidean space R³ is a separable metric space. true or false?

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…

Q: Prove the following: Let (Y, d) be a metric space. Let B be a bounded subset of Y. If A C B, then A…

A: Let Y, d be a metric space and B be a bounded subset of Y. Suppose A⊂B. To prove that A is bounded…

Q: Prove that: 1. (R, J. |) is a complete metric space. 2. (Q, I. |) is not complete metric

A: To prove that R,. is a complete metric space.  Suppose that xn is increasing and bounded, Let…