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: Exercise. Is it possible for a metric space (X, p) to contain more than one point and be such that…

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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 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 Y be an infinite subset of a compact set X c R. Prove Y' # Ø.

A: Y' denotes the derived set of Y i.e. the set of all limit points of Y. We just need to show that any…

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: Let (X, d) be a metric space and let A ⊆ X be complete. Show that A is closed.

A: Given that X,d be a metric space and A⊆X be complete. The objective is to show that A is closed.

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: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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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: 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 Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

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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: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

A: Given: A subset (a, b)⊆ℝ is T2-space in ℝ, Tu. To determine: Whether given statement is true or…

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

A: Solution

Q: Let X = {a,b,c} prove that topological space on the set X T = {X. Ø, {b}, {a,b}} be a

A: Let X={a,b,c} be a set. To show that T={X,ϕ,{b},{a,b}} be a topological space on the set X.

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

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Q: Let (X, M, u) be a measure space wE

A: Given, (X, Μ, μ) be a measure space where μ is a probability measure. Assume thatf, g : X→ℝ are…

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

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Q: A of a metric space (x, d.)

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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: 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: 3. Let a, b e R such that a < b. Prove that the open interval (a, b) is an open set in R.

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

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Q: Prove that the open ball B(p, 8) = {x : ||x – p|| < 8} is a convex set. [Hint: Use the Triangle…

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Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

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: 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: 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: (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