(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 both open and

Q: j) A U A = A k) A U $ = A

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Q: Let A be a set. Suppose that f: w→ A is onto A. Prove that A is countable.

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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: To prove the theorem that states that “S” is a countable set, and there exists a surjection ofℕ onto…

Q: Let S be a nonempty subset in R". Prove or disprove by a counterexample that Span S = (S+)+

A: We have to prove or disprove by counterexample that Span S=S⊥⊥

Q: Prove that there is no set to which every se Show that for any nonempty set A, there xE Bex belongs…

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

Q: 2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p…

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Q: Let x E A. Prove that A is an infinite set if and only if A≈ (A - {x}).

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Q: Let S and T be sets. Prove that SUT = T iff S is a subset of T.

A: To Prove S∪T=T iff S is a subset of T

Q: Prove that the set S = {n² : n e N} is denumerable. %3D

A: We have to show that the given set S is a denumerable set.

Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: Let A be a compact set in R. Show that the complement A is open and contains an interval of the form…

A: Solution: Before going into the problem we must know the Theoem named Heine-Borel Theorem.…

Q: Let S be a non-empty bounded subset of R and let a be an upper bound of S. Show hat sup S = a if and…

A: Follow the steps.

Q: 2. Let S is a subset of R. Prove that S is closed if and only if S contains all of its accumula-…

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Q: 2) L Let A and B be di screte sets of R". Def ine as p(t )= (1-t )a + tb for atA and b€B. tER,…

A: Given: A and B be discrete sets of Rn. Also, it is defined as: Pt=1-ta+tb for t∈R. And, a∈A , b∈B.…

Q: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

Q: . Let S be a bounded set in R and let T be a nonempty subset of S. Show that inf S <inf T< sup T <…

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Q: Prove that: m*(A) 2 0, for all sets A

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Q: . Let A CQ satisfying the following conditions: (a) A + Ø and A # Q. (b) If p E A and q E Q with q <…

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Q: (a) Investigate whether d: RxRR delined as d(z, u) = (- u) is a metrie on the set ol all real…

A: We investigate given distance function is metric on R or not

Q: Let S be a non empty subset of R and let u e R be such that u + is an upper bound of S for each n e…

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Q: [Used in Example 1. 6. 3] Let T be the set defined in Equation 1. 6. 1. (2)Prove that if T = D, for…

A: Given: The set                                                                   T=x∈ℚ | x&gt;0 and…

Q: 2) Let A be a finite set and a EA.Lf a EA is not a maximal element of A, then a has an îmmediate SUC…

A: Given A is a finite set. a∈A is not a maximal element of A. Now we have to prove 'a' has an…

Q: Q3. Let N = {1, 2, 3, ..} equipped with the cofinite topology. Prove that M = {7,14, 21} is a closed…

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Q: Let T and M be a non-empty families of sets with T  ⊆ M. Show that ∩T  ⊆ ∪ M.

A: Given: T and M are families of sets with T⊆M. To show: ∩T ⊆∪M

Q: Suppose that |x| = n. Find a necessary and sufficient condition on r and s such that (x") C (x*).

A: Given,

Q: Consider the subset W = {x € Q : xª > 1} of Q. 2.1 Express W using interval notation. 2.2 Is W…

A: W = x∈ℚ: x4&gt;1 This implies that W is a set of rational numbers 'x', such that x4&gt;1. if x4…

Q: 1) Let X be any non-empty set and xo E X. a) Prove that T = {X}U{AC X : xo ¢ A} is a topology on X.…

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Q: Let A be a finite set, and suppose that f : A → A is one-to-one. Does f necessarily have to be onto?…

A: The given question is solved below.

Q: Prove that P(Sn T) P(S) n P(T) for any sets S and T

A: Let, S and T be a any set. also, P(S) be any power set of S and P(T) be any power set of T.x∈P(S∩T)…

Q: prove that the outer measure of Cantor Set C (x), 0<x< 1, is 1-x the generalized

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Q: Prove that if x is not free in the formula A, then ` A ≡ (∃x)A

A: The formula is -------(1). To Prove: x is not free in the formula Proof: Let us assume that A(x) :…

Q: For any set A and any ≥0, there exists an open set O such that A CO and m* (0) ≤m* (A) + €.

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Q: Prove that if R, S, and T are nonempty sets, then R×T ⊆ S×T if and only if R ⊆ S.

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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.

Q: Show that Let ECR" is measurable set. If µ(E) >0, then E have a non-measurable subset

A: Here, we introduce the notation Gr :=G∩[0, r) + 1-r ∪ G∩[r,1) - r for G⊂[0,1) and r∈R. Note that…

Q: Q3. Let N = {1, 2, 3, ...} equipped with the cofinite topology. Prove that M = {7,14, 21} is a…

A: Given below the detailed solution

Q: 3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a)…

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Q: Let E and F be measurable sets each having finite measure. Prove that m(EAF) = 0 if and only if m(E)…

A: To prove: The statement "Let E and F be measurable sets each having finite measure. Then, mE∆F=0 if…

Q: 2. Let f : X → Y be onto. For each b e Y, let A, {As : b € Y} is a partition of X. f6]. Prove that

A: Given that f:X→Y is an onto function. Then for every b∈Y there exists x∈X such that  fx=y Also,…

Q: 1. Let T and S be topologies in X. Show that SnT is also a topology in X.

A: 1.

Q: 1. Consider the collection M = {0, X}. Obviously, X E M. Note that X = 0 is in M. Since M only…

A: Suppose A be any set. Then, Power set of A is defined and denoted as:P(A)={B:B is a subset of A}

Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

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Q: Let A and B be nonempty subsets of R such that A is bounded above and B is bounded below and let A –…

A: Let A and B are two non-empty sets of ℝ.Given A is bounded above and B are bounded below sets.Claim:…

Q: Let f : (X,d) - (Y, d1) becontinuous fumction, then for any closed set G in X, f(G) is not necessary…

A: False

Q: 1) Let X be any non-empty set and xo E X. a) Prove that T = {X}U{ACX: 2o ¢ A} is a topology on X. b)…

A: Topology means we have to prove 3 conditions then we can tell that space is Topology.

Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤…

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Q: 1. Let X be any nonempty set and let p E X, Prove that : E, {G X:p¢ G}U{X} is a topology on X.

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

Q: 5. Let S be a nonempty subset of R that is bounded above, with upper bound b. Prove that b = sup(S)…

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Q: Let S be a bounded set in R and. let a > 0, and let aS := {as : s E S}. Prove that: i. inf(as) = a…

A: