The only subsets of R which are both open and closed under the discrete metric are Ø and R. Select one: O True O False

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: Let X = [0, 1] U [2, 3] endowed with the standard metric. Then [0, 1] is an open subset of X. Select…

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Q: Prove that in a metric space (S,d) every closed ball Sr[Xo] is a closed set

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Q: If there exists open ball B (a, r) such that B (a, r) NG= Ø then a is not limit point of G True O…

A: TRUE

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

A: In mathematics, the closure of a subset S of points in a topological space consists of all points in…

Q: (b) Prove that the set S = {x EQ: √2<x<√3} is closed and bounded in (Q.1-1), but not compact in…

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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: A subset I of a metric space R with the usual metric is compact if and if only it is an interval…

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Q: In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and…

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Q: Let d and e be metrics on a set X such that for each ball Ba center at pE X there exists a ball Be…

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Q: Let (X.d) be a metric space, x ∈ X, epsilon > 0, and E = { y∈ X: d(x,y) ≤ epsilon}. Show that E is…

A: A set E is closed in X if it contains all of its limit points.

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

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Q: Let

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

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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: Exercise 5. 1) Show that the metrics dp, p ≥ 1 and dmax on Rª are all Lipschitz equivalent. 2) If X…

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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 be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: The objective is to choose the correct option: Let X be a discrete space then  a) X is homeomorphic…

Q: If there exists open ball B (a, r) such that B (a, r) nG=Ø then a is not limit point of G

A: To check whether there exists open ball B(a, r) such that B(a, r) ∩ G= ∅, then whether a is not…

Q: 4. Let (X, d) be a metric space and xnx where each x, e X and x e X. Let A be the subset of X which…

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Q: Prove the following sets are not compact by finding an open cover that does not have a finite…

A: a) Given-  A set A=−1,1. Explanation- Consider a set Xn=-2,1-1n for n=1,2.... Then clearly,…

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

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Q: Let X = R, and d be the discrete metric, and A = [3, 6) Show that if A is compact or not

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

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Q: (1) Determine if each of the following statements is true(T) or false(F). 1- In a metric space,…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

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: Let x be an interior point of a subset S of a metric space (X,d). Show that x must also be a limit…

A: Interior Point: Let A be a subset of metric space X, d. A point x∈A is called an interior point of A…

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: The set S=xER:x²-4<0} with the usual metric is . O A. Compact. B. connected. C. Not connected D.…

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Q: Let A CX with the discrete topology. Prove that ô(A) = Ø.

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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: Given the discrete metric space for n E Na the discrete metric o defined by the distance function 0…

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

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Q: consider R with the usual topology then a subset A={1,0.5,..} is: empty or closed or open or none…

A: Given R with the usual topology and given a subset  A=1, 12, 13,.....

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: Prove that Let X be a metric space such that X E1 U E2. If E1, E2 are complete, then X is 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 be a compact subset of R and let C(X) denote the set of continuous real-valued functions on X…

A: let X be a compact subset of ℝ and let CX denote the set of continuous real valued functions on X.…

Q: The set E={xER:xs2x² – 1} with the usual metric is compact.

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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: Let A be a subset of R with the usual metric such that A=A. If there exists r>0 such that d(x,y) <r…

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Q: Let X= (3, 5] u {6,7,...} and A= {5,6,7} and metric defined on X as d(x,y) = |x - y|

A: Note: As per bartleby instruction when more then three questions for subpart is given only three…

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

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