6. Let A and B be bounded nonempty subsets of R, and let A+ B:= {a +b: a e A, be B). Prove that sup{A+ B) sup A + sup B and inf(A + B) = inf A + inf B. %3D

Q: Show that if A is such that m (A)> 0, then there exists a bounded subset BCA such that m'(B) > 0.

A: By properties of Measurable set,         If then  .

Q: 21. Let A and B be two on-empty bounded sets of positive embers and let С%3D {ху: х€ А and y € B}.…

A:

Q: Suppose that A and B are subsets of R which are nonempty and bounded above, and suppose that BCA.…

A:

Q: Assume that A and B are nonempty, bounded above, and satisfy B CACR. Show that sup B < sup A.

A: Introduction: The supremum and infimum properties are the very important properties of any subset of…

Q: We define the included point topology by Tp={ UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in Rif…

A:

Q: 2. if it was P(x)=ao+a,%tazk Prove that PecR) inner multiPlication sPace

A:

Q: 1. For a subset YC (X,T ) , show that the collection TY ={Y NUJU ET} is closed under finite…

A:

Q: Suppose A ⊂ R is both closed and bounded, prove that A has the Heine-Borel Property

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Q: 1. Prove by Induction that the numbr f edges of Kn

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Q: 2. Consider the set A = { n+m² n2+m m eN}. : п, тEN (a) Prove that sup(A) = ∞. (b) Prove that A is…

A: Set and infimum value

Q: Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.

A:

Q: Let (X, E, µ) be a measure space. Justify through an example that the hypothesis µ(X) < o in…

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Q: Show that if A is closed in Y and Y is closed in X, then A is closed in X.

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Q: Let S be a nonempty bounded subset of R and let m = sup S. Prove that m is in S iff m = maxS

A: According to the given information, it is required to prove that:

Q: We define the included point topology by Tp={ UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R…

A: option (d) is correct.

Q: F, there is a disconnection A, B of F such that x E A and y E B. 8.X. If C1, C2 are connected…

A: We use the fact that a set X is connected if there is no continuous function form f from X to Y=…

Q: Show that the set (1) isopen u k = 1 and find its measure.

A:

Q: Given a set L sR", Following are true statements regarding conv L(convex hull of L):

A: In this question, we have to choose the correct option for the convex hull. the set L⊂Rn By using…

Q: 5. Show that if A and B are bounded subsets of R, then A U B is a bounded set. Show that sup(A U B)…

A:

Q: Show that a set {V1, ..., Vp} in R" is affinely dependent when p > n + 2.

A:

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 <…

A:

Q: Show that if A and B are bounded subsets of IR, then inf(A U B) = min{inf A.inf B}. %3D

A: To show  If A and B are bounded subsets of R then inf(AUB)=min{infA .infB}

Q: We define the included point topology by Tp-{UcR;U=Ø or pEU). Let A [3,5L then A is dense in R if *…

A:

Q: We define the included point topology by Tp-{UCR;U=Ø or peU). Let A = [3,5[, then A is dense in R if…

A:

Q: We define the included point topology by Tp={ UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R…

A: R is equipped with Tp and p=3

Q: Show that if A is closed in X and B is closed in Y then A × B is closed in X × Y

A:

Q: We define the included point topology by Tp={UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R if…

A: The detailed explanation is given below:

Q: Let A and B be bounded nonempty subsets of R, and let A + B:= {a +b: a E A, be B). Prove that sup(A…

A:

Q: Let A, B Be two subsets af R, such that B is bounded and YaE A ]bEB such that show that: a sb O A is…

A:

Q: The set S=[1,3) is not compact. Which open cover of S contains a finite subcover?

A: This is the question of Topology.

Q: Let S be a non-empty open subset of R that is bounded above. Show that sup S exists but that max S…

A:

Q: If A and B are nonempty bounded subsets of R with A C B, prove that sup A < sup B.

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Q: If A and B are bounded subsets of R, then AuRand AOR are bounded: prove that

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Q: 5. Suppose that A and B are bounded subsets of R. Prove that AUB is bounded and that sup(AUB) =…

A: Given that A and B are bounded subsets of R.   We have to Prove that AUB is bounded and that…

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

A:

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

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Q: Consider a finite set S of points in the two-dimensional plane. A point p of S is called a covidiot,…

A:

Q: • Show that if A is compact and Bis closed, and d(A, B) = 0, then ANB÷Ø.

A:

Q: Let A, B be disjoint infinite sets. Prove that their union A U B is denumerable if and only if both…

A: Given A , B be disjoint infinite sets. Firstly, consider A∪B is denumerable Since, A⊆A∪BandB⊆A∪B…

Q: Consider X = {a1, a2, ..., an} Can a topology of X with 3 open and one with 4 open be equivalent?

A: We have to verify the given statement.

Q: Let the non-empty set S = {x - yv3:x,y ER}, show that S is closed'under multiplication then find the…

A: We have to show that the given set S=x-y3:x,y∈R is closed under multiplication and also find the…

Q: a nonempty compact set of IR. sup Ke K. be

A:

Q: 2

A:

Q: Suppose A is a nonempty subset of R. (a) Is it true that Là=L? Prove or disprove. (b) Let iso(A)…

A: Given that A is a non-empty subset of ℝ. (a) To prove that                      LA¯ = LA

Q: Let D be the closed region in

A: Given,                    x2-2xy+y2+x+y=0   and      x+y+2=0

Q: We define the included point topology by Tp={ UcR;U=Ø or peU}. Let A = [3,5[, then A is dense in R…

A:

Q: Prove that S UT = SN T for all sets S and T using the rules for complements of sets.

A: rules of complements x∈A⇔x∉A¯

Q: Let A = {x E Z* : x is a mutiple of 2 and x < 15}. Rewrite A in roster notatlon.

A: Given that : A = {x∈Z+ : x is a multiple of 2 and x &lt;15 }   Roster form : A representation of a…