Let A1,·.., Am be compact subsets of R. Prove that Aı × · … × Am is a compact subset of Rm.

Q: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

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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: Consider the following statements: (i) If A is compact, then both sup(A) and inf(A) exist and are…

A: Statement (i):-If A is compact, then both sup(A) and inf(A) exist and are elements of A Proof :- if…

Q: LAUBUC = ANĒNT. %3D

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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 a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

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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: For any collection a of subsets of X, there i ntaining a.

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Q: Prove that the class {AC – B, BC – A, AB − C} is pairwise disjoint.

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Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U ·. U Bn equal A exactly?…

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Q: Prove that the complement of a dense open subset of R is nowhere dense in R

A: Prove that the complement of a dense open subset of R is nowhere dense in R

Q: If A is a subset of B and P(B)=0,then prove that  P(A)=0.

A: Assume that A is a subset of B and P(B)=0.

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: required to prove that A×∅=∅

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

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Q: Show that the set K={a, b} with actions + and - defined tables   is a body whose zero element is a…

A: We are the set K={a, b} with actions + and ∙ defined in the following tables. Notice + operator…

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: 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…

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Q: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U... U Bn equal A exactly?…

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Q: Let A = {AC R| A is countable} and A2 = {A C RI A° is countable} which is a measurable space? (IR,…

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Q: 6. Let A and B be bounded nonempty subsets of R, and let A+ B:= {a +b: a e A, be B). Prove that…

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Q: If U is homeomorphic to V and U is complete, then" O V is compact O v is not necessarily complete O…

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Q: Let τs and τ be the standard topology and the countable complement topology on R, respectively.…

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Q: Given an infinite set X, construct a bijection from X to a proper subset of X.

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Q: (a) Let A be a subset of R". Show that (A = (A°)°.

A: WE proved it using Claim 1 and Claim 2.

Q: compact, then E must be infinite. Exercise 6.46. Prove that if E is a closed subset of a compact set…

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Q: Prove: If U is a neighborhood of x0 and U ⊂ V, then V is a neighborhood of x0. Also, if U1 ..., Un…

A: The neighborhood of a point: Let (X, T) be a topological space and x0∈X. A neighborhood of x0 at is…

Q: Let A be a subset of some universe U. Prove that AN A° = Ø.

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Q: (c) Prove that only Ø and R are open subsets of R

A: A subset Uof a metric space (M, d) is called open if, for any point x in U, there exists a real…

Q: Prove that if Gn is an open set which is dense in the complete metric space (S,d) for n = 1,2,...…

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Q: Let (E), be a countable disjoint collection of measurable sets. Prove that for any set A, 00 m* (AnU…

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Q: Prove A is a subset of (A U B)

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Q: Let {Aα:α∈I}{Aα:α∈I} be a nonempty indexed collection of sets.  Prove or disprove…

A: Let Aα:α∈I be a non empty indexed collection of sets.  If A is a family of sets, the intersection…

Q: Prove that if A Cw is nonempty and UA = A, then A = w.

A: Given that, A⊆ω is non empty. And ∪A=A. To show A=ω it is enough to show that,  A⊆ω and  ω⊆A. One is…

Q: 2. Prove thatN is the smallest inductive subset of R:

A: We can prove this by proving property of induction

Q: Let A and B be two subsets of some universe U. Prove that (ANB)° = A°U B°.

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Q: Prove, without using Theorem 1.12 or copying the proof, that for all a, b ∈ Q with a < b, there…

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Q: Let A, B, C be sets. Prove that if f: A → B and g: B → C are bijections, the composition gof: A → C…

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

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

Q: (b) Prove that the complement of a dense open subset of R is nowhere dense in R..

A: Prove that the complement of a dense open subset of R is nowhere dense in R.

Q: Show that there must be two distinct subsets A, BC S such that

A: Every value in S is between 0 and 100 , So the sum of any subset will be at least 0 and  at most…

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: the collection of all close subsets of R is called the topology of R ylgn İhi

A: False

Q: Prove that if A is closed and B is open, then A \ B is closed.

A: Solution:-

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…

Q: Let E1, E2 be two compact subsets of RP. Use the Heine-Borel Theorem to prove that EU É, is also a…

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Q: Prove that only ∅ and R are open subsets of R.

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Q: If K is a compact subset of RP and z is a point of RP, then the set K, = {r+y :y € K} is also…

A: Given that K is a compact subset of ℝp and x is a point of ℝp. The objective is to show that…