Remark. For everyr E N, a set in R" with the Euclidean metric is com- pact if it is bounded and closed. However, in arbitrary metric spaces, boundedness and closedness do not necessarily imply compactness. Exercise. Give an example that satisfies the remark above?

Q: 3.2 Prove that in any metric space (S, d) every closed ball S,xo is a closed set.

A: Let (S, d) be a metric space. Consider a closed ball Srx0 with both center x0∈S and radius r>0…

Q: 5. If G1,G2 be two given closed sets in a metric space (X, d) prove that: G1 U G2 is closed.

A: We have to prove given result:

Q: Let A and B be subsets of topological spaces X and Y, respectively. Prove that CI(A × B) Cl(A) ×…

A:

Q: Suppose A and B be subsets of metric space. Show that if A is compact and B is closed then A + B is…

A: Given A and B are subsets of a metric space. and also Given A is compact , and B is closed. We have…

Q: Show with all the relevant details, albeit succinctly that every compact metric space M is complete.

A: Here we use sequential definition of compactness so that we use that in proving the completeness of…

Q: 2. Let E C X where X is a metric space. Let E° denote all the interior points of E. (E°is an open…

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Q: A metric space (X , d) and a collection of open sets U such that UEU is not open.

A: Let X, d be a metric space. Let U be the collection of open sets in the metric space X, d. That is,…

Q: State whether the following is true or false: a) The set [0,4] with the usual metric is…

A: (a) The set [0,4] with the usual metric is sequentially compact. This statement is true. Considering…

Q: Show that an infinite intersection of closed sets F, k = 1, 2, 3, . .., in a metric space (S, d) is…

A: first we will proof that an infinite union of Open sets in metric space ( S, d) is open using the…

Q: In a metric space (M, d), show that both Ø and M are open sets in M. In a metric space (M, d), show…

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

A:

Q: 3.2 Prove that in any metric space (S, d) every closed ball S,ro] is a closed set.

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Q: let and be nonempty metric Space x E X. Ihen A B Sebsets of and let d(x, A), dlA, B) and dlA) are…

A: As per bartleby guidelines for more than three subparts in a question only first three should be…

Q: Use the definition of compactness to prove that a closed subset Y of a compact set X ⊆ R is compact

A: In this question, we use the definition of compactness to prove that a closed subset Y of a compact…

Q: Exercise. Every point in a discrete metric space forms an open ball.

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Q: Which of the following statements is not true? (i) In any metric space (X,d) every closed ball is a…

A:

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,…

A:

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

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Q: The set X is an open set in the ambient metric space . O True O False

A: Solution

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: Which one of the following statements is true? * O Rwith the Euclidean topology and R with the…

A: Which of the following statement is true?

Q: Which one of the following statements is true? O R with the Euclidean topology and R with the…

A:

Q: Given a Euclidean space E, if f: E→E is any self fanguidodge door of pas food of go

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Q: Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact

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Q: Which of the following are compact spaces? (a) S" with a finite number of points removed. (b) The…

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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: 3. Let K,and K, be compact subsets of a metric space X. Show K, U K2 is compact

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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: None of the choices. Consider the following subsets of R equipped with the Euclidean topology: * Зт…

A:

Q: Let (M, d) be a metric space and K ⊂ M. The following statements are equivalent: (i) K is relatively…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Which one of the following statements is true? * R with the Euclidean topology and R with O the…

A: Any two homeomorphic spaces always shares the same Topological property. This means if one of them…

Q: 6. Let E CX where X is a metric space. Show the limit points of E are the same as the limit points…

A: Limit Point: Let be a metric space. A point is a limit point of a set if for each there exists an…

Q: ) Determine if each of the following statements is true(T) or false(F). 4- For any closed subset A…

A: There are given three statements we have to find are they true or false

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: the following subsets in an indicated metric space determine the interior, boundary points and…

A: Ans:

Q: The set {0,3} with the usual metric is connected as union of connected sets {0} and {3} True False

A:

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: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.

Q: Task 13 Let (M, d) be a compact metric space. Show that closed subsets of M are compact.

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Q: Prove that in a metric space (X, d) every closed ball that is a set K(x, r) = {y e X : d(x, y) <=…

A: Question: Prove that in a metric space X, d every closed ball K¯x, r=y∈X, dx, y≤r is a closed set.…

Q: For any metric space (X, d) and x EX, the singleton set (x) is closed. Select one: OTrue O False

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Q: State whether each of the following statements is true or false, in either case…

A: Solution of the problem as follows

Q: c) Let (M, d) be a metric space and let S,T≤ M. Prove that SOT = ST. But SOT SOT.

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Q: 3. Consider the metric space (R..|) and let S be the union given by (0}U UI J. Note that each of the…

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Q: 5. State whether the following is true or false m) A subset I of a metric space R with the usual…

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Q: Which one of the following statements is true?* R with the Euclidean topology and R with the…

A: Which one of the following statement is true

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

A: Here given usual metric on the set of real numbers R

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

A: