Let A be a non-empty subset of the metric space (X,d). show that a. d(x, A) – d(y, A)| < d(x, y)For all x, y E X With d(x, A) = inf{d(x, y): y E A} %3D

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Q: (2.3.15. Let X be a metric space. (a) Prove that SCX is bounded if and only if diam(S) < 0o.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = A n (A)° = A – int A…

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Q: Let (X, B(R), ft) be a Borel measure space. ANC= 6 BOC =0 p(AnB) = 0. Determine ju(AUBUC) %3D

A: Since sets are disjoint so measure will be the sum

Q: Let A be a non-empty subset of the metric space (X,d). show that |d(x, A)- d(y, A)| < d(x, y)For all…

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Q: Let (X, d) be a metric space, x E X, e > 0, and E = {y Ex: d(x, y) ≤ e}. Prove that E is closed.

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Q: 2. Prove that if d is a metric for X, then another metric d' for X is given by if a ≤ b " if b ≤ a…

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Q: Q. Let (X, B(R), µ) be a Borel measure space. f AnC= φ, Β nC = φ, μ (Α n Β) -0 Determine µ(AUBUC).…

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Q: . Let X be a nonempty set with metric d. Show that the function e defined by e(x, y) = min[1, d(x,…

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Q: Define on R, the metric d defined by d(x, y) = |1+1리 1+ |y| Then (R, d) is not complete. True False

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Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = A O (A)° = A – int A…

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A: Yes, it is always true. See the proof.

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