Exercise. Is it possible for a metric space (X, p) to contain more than one point and be such that the only open sets are 0 and X? Justify your assertion.

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

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Q: Which sets are open in the metric space (IR, d), v i). [1, 2) ii). (1, 2) U (2, 3)

A: ii). (1, 2) U (2, 3)

Q: Let (N, d) be the usual metric space, then the set A= {1,3} is O closed only O neither open nor…

A: ( N,d) be a usual metric space of natural number.  A={1,3} be a finite subset of N. Since A is…

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

A: Answer

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: Let X be a topological space. Prove that (a) Ø and X are closed. (b) Intersections of any number of…

A: Union of any open sets is open and intersection of two open sets is open.

Q: Let (X,t) be a topological space, A, B C X. Then A' UB' = (A U B)', A'n B' c (A N B)' Select one: O…

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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: Let

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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…

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Q: If X is a normed space then the open ball and closed ball is a Example: convex set

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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: Let A=(0, π ) υ (π e?] %3D c. Is A a closed set? d. Does A contain any isolated points? е. Find the…

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Q: Exercise. Prove that in a metric space (X, p), a set G C X is open if and only if it is a union of…

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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: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

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: Define the Cartesian product of sets X and Y? How is this Cartesian product denoted?

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

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Q: Exercise 2. 1) If (X, d) is a metric space show that B(x,r) is always an open set for x = X and r≥…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

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: 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: Exercise. Prove that in a metric space (X,p), a set G CX is open if and only if it is a union of…

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Q: Exercise 8. Prove the following properties of neighborhoods: Proposition. If x is a point of the…

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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: Prove that in any metric space, an ɛ- eighborhood of a point is an open set.

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

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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: (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: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

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

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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: 2- In a metric space (X, d), A is the intersection of all closed subsets of X which contains A.

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Q: Exercise 60 Let p and q be two points in a topological space X. Show that there exists an open set U…

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Q: Is any discrete space complete? Give an example to substantiate your answer.

A: A metric space is complete if and only if every Cauchy sequence is convergent.

Q: Exercises: 1) In a metric space (M,d). show that both Ø and M are open sets in M.

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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: Exercise. Prove that in a metric space (X, p), a set GC X is open if and only if it is a union of…

A: Solution

Q: If A is a compact subset of a metric space (X, d) and B is a closed subset of A, prove that B is…

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