2. Prove B = { :n e N} U {0} is compact. %3D 3. Let F = {Fa : a € A} be a family of compact sets. Prove O Fa is compact.

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Q: (6) Let (X,M.) be a measure space, and {Ej}cM Prove that : (I) If E,SE,S…. ,then j (II) If E,2E2...…

A: The given problem is related with measure theory. Given that X , M , μ is a measure space and Ej1∞ ⊆…

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A: Given:

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Q: (a) Let S {2 ringleton is open set in (M, d). oce and let ACM. Prove that aA = O if and only if A is…

A: The problem is from Topology.

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Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

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Q: Q: In usual topological space (R.Tu). answer the following: 1- Is [0,1] is compact set in R? Why?…

A: Note: Hi! Thank you for the question, as per the honor code, we are allowed to answer three…

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Q: Show that the sete S = {x E R]x = m/3, where m E N with m > 4} is countable

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Q: 2. Show that the set 3 S = 3 2 2 2 2 spans all of R³ or find an element of R' not belonging to…

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Q: 1) In a metric space (M, d), show that both 0 and M are open sets in M. 2) In a metric space (M, d),…

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A: Please find answer below:

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Q: 2.12 let A and B bounded sets For which there is an ε >o such that la-blze for all ac A, be B .…

A: Consider A and B are bounded sets.

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Q: 5-The Center of aringR dented by cent CR)is the set Cent CR) - [CERIC.X = X-C,V XER? %3D Prove that…

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A: As per our guidelines we are supposed to answer only one question.Since you have asked multiple…

Q: let A Set bounded below, ילC erove that C. nf A inf C A %3D

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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.

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A: Give an example of a collection of compact subsets in R, say (Kn) n=1 to infinity,  such that U n=1…

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Q: let E be a set that baunded. Prove that Inf E = Sup -E) where R, XE E y

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Q: Suppose that a,beG such that a = m and b =n with Gcd (m,n) = 1. Then n | is in mn

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Q: 3.3 Let 11 and ra be distinct points in the metric space (S, d). Verify that there are open balls…

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