Give an example of a collection of compact subsets in R, say (Kn) n=1 to infinity,  such that U n=1 to infinity Kn is not compact.

Q: If A and B are bounded subsets of R, then prove that AUB and AB are bounded.

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Q: 2. Let S be a bounded set in R and. let a < 0. and let aS := {as : s E S). Prove that inf(as) = a…

A: Let S be a bounded set in R and let a&lt;0. And let aS := {as : s belongs to S}. We have to prove…

Q: Every closed and bounded subset of X is compa The subset {z EX: |||| ≤ 1} of X is compact... X is…

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Q: Let A be a non-empty subset ofR which is bounded below and let B be the subset of R defined by B =…

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Q: In an indiscrete space X * O None of the choices O the only dense subset of X is X every subset of X…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: (ii). Prove that a set S CR is everywhere dense iff it intersects every non-empty open sets in…

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

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Q: Prove that the set of integers is nowhere dense in R.

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Q: Prove that intersection of any class of convex subsets of N is convex?

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Q: A and B are bounded subsets of R, then prove that UB and AnB are bounded.

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Q: In a discrete space X * the only dense subset of X is X every subset of X is dense in X None of the…

A: In topology , a subset A of a topological space X is called dense (in X) if every point x in X…

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: 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: Let W be a nonempty subset of R that is bounded above, and define the set B as follows B= {N ER:N is…

A: Let W be a non-empty subset of ℝ is bounded above, and define the set B as follows, B=Ω∈ℝ: Ω is an…

Q: Let A be a denumer able set. Prove that A has a denumer able subset B such that A - B is…

A: Let A =set of multiples of 4, B=set of multiples of 3. Both are denumerable. Intersection of A and B…

Q: 3. Prove that if G is an open set dense in the metric space (S, d) then SG is nowhere dense in S.

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Q: prove tuat tue Complement of a d sea dense open subset Of 1R is nowhere dense in IR. N-B Please…

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Q: Prove that if A is a nonempty finite set and B is denumerable, then A × B is denumerable.

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

A: Given, A1,A2, ... , Am are compact subsets of R. We prove A1x A2 x ... x Am is a compact subsets of…

Q: Q2: Let the set under consideration be N for each n EN, define u, = {n, n+1, n+2, ...} and let 1 =…

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Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

Q: (ii). Prove that a set SCRis everywhere dense iff it intersects every non-empty open sets in R.Prove…

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Q: 2. Prove that if K is compact in R and F is a closed subset of K, then F is compact using (a) the…

A: To Prove- Prove that if K is compact in R and F is a closed subset of K, then F is compact using…

Q: Show that the given collection F is an open cover for S such that it does not contain a finite…

A: a cover of a set {\displaystyle X} is a collection of sets whose union includes {\displaystyle X} as…

Q: ã) In general, for any metric space (M, d), we have 1) Sis compact in M = S is closed and bounded in…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

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

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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: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

Q: 1.Any open interval in R is an Open set in R 2.The {0} is open in R with discrete metric

A: We will use the fact of open ball

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

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Q: 9.C. Prove directly that if K is compact in Rº and F is a closed subset of K, then F is compact in…

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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: Prove the following. 1. Bolzano-Weierstrass Theorem: If a bounded set S in R" contains infinitely…

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Q: Suppose f(z) is holomorphic in an open connected set N. Can g(z) = f(z) be holomorphic in N? Explain…

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Q: Let {An : n E N} be the indexed collection of sets defined by 1 2, 2+ 2n An Prove that: nMeN An = Ø

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Q: 3. Let E be a nonempty subset of R that is bounded above, and set U = {ß ER is an upper bound of E}.…

A: Given that Let E be a nonempty subset of R that is bounded above, and set U = { B belongs to R is an…

Q: Prove that a set A is countable if and only if there exist an injection of A into N.

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Q: show by an example that a nonempty proper subset of S in the metric space (S, d) can be both open…

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Q: Find an example of a nonempty set satisfying: *Not Open, Not Closed, Dense in R *Open, nowhere…

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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: 3. Prove that if K1 and K2 are compact sets in R, then K1UK2 is compact in R using (a) the…

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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: Prove that if A and B are disjoint denumerable sets, then A UB is denumerable.

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Q: In a discrete space X O every subset of X is dense in X the only dense subset of X is X O a proper…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Consider the subset G {r• /2: x € Z} of R. Show that G is closed un- der addition.

A: The given subset is: G= x·2 : x∈ℤ

Q: A metric space is said to be separable if it contains a countable dense subset. (a) Is R separable?…

A: Set of all rational numbers Q belongs to R and Q is dense in R . Also set of all rational numbers Q…

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: Find an infinite collection {Kn: n N} of compact sets in R such that the union U K₂ is not compact.…

A: We know the definition of compact sets , A subset S of ℝ is said to be compact if every open cover…