Let ACR be a nonempty bounded set. Prove that sup A is unique.

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…

A:

Q: Show in the discrete metric space: 1. All sets are open (especially single points);

A: Let X be a non-empty set and d be discrete metric on X

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

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

A:

Q: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

Q: let A be a bounded. Sup A non. n-empty set in R show that is a limit point of A.

A:

Q: Suppose that A and B are bounded, nonempty subsets of R. Prove that AU B is bounded and that sup(AU…

A: Consider the provided question, Suppose A and B are bounded, nonempty subsets of R.

Q: If S is compact and D is in S is a closed set, then D is compact. Use the definition of…

A: Let S is compact and D be a closed subset of S. We are to prove that the set D is compact in S. Now…

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

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

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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 x be an infinite set and let ta be a topology on x in which all infinite subsets of x are open…

A: Given that X be an infinite set and let τ be a topology on X in which all infinite subsets of X are…

Q: ] Prove that a nonempty closed subset of R, if it is bounded from below, has a least element.

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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: prove that an is bounded above.

A: In this question, we prove the sequence  an=(1+1n)n is convergent by given porcedure. i.e.…

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: Use the Heine-Borel theorem to give an example of a set not compact in R with the usual topology…

A: An example of a set not compact in R with the usual topology and a non-compact set R2 with the usual…

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 X be an infinite set T be topology of the a Linfinite subsets that is on x- and let on X in…

A: Given : X be an infinite set and T be a topology on X such that all infinite subsets of X are open.…

Q: Every connected subset of R is complete is ture or false?

A: Every connected subset of R is complete is true or false?

Q: Exercise 6.40. Let E be a nonempty subset of R that is bounded above. Prove that if a = sup E, then…

A: This is the question of topology.

Q: rcise: Prove that every finite set S of R" has no accumulation point.

A: Proof of the the given theorem is given as

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: Prove that for any infinite index set J, the uniform topology on R° is strictly finer than the…

A:

Q: Let A be a nonempty subset of R that is bounded below. Define the set B = {b: b is a lower bound of…

A:

Q: Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact. * O…

A: "Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact." The…

Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

Q: Prove that every finite set S of R" has no accumulation point.

A: Now taking set S Let ? = {?1, ?2, . . . , ?n }.

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 nonempty bounded subsets of R with A C B, prove that sup A < sup B.

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Q: Let A, BCR be two non-empty bounded sets. Assume that for every x E A, there exists y E B such that…

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Q: In R with the cofinite topology, every preopen set is open

A: As we know that , In ℝ with discrete topology of every semi open set is open.

Q: Question No 1 (8). Prove that the discrete topology on an uncountable set does not Satisty the…

A: Note- Let X be a topological space, then X satisfies the second axiom of countability or is second…

Q: Give an example of a closed subset S of R2 such that conv S is not closed.

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Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

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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: Let ||.|| and ||..|| be two norms defined on space X.Show that ||.|| and ||..|| are equivalent if…

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Q: 4. Use the least upper bound property of R to prove that every nonempty subset of R that is bounded…

A: Solution

Q: Show that all open balls are bounded subsets of a metric space.

A: Solution: Let M=X,d be a metric space. For an arbitrary point a∈X, the open ball Bεa where ε&gt;0 is…

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: Let (X,τ) is a topological space and A ⊆ X. If all subsets of A are closed in X, then set A cannot…

A: Given that X,τ is a topological space. Let's prove this theorem by contradiction. So  X,τ is a…

Q: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

A: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

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

A:

Q: ndary of a path-connected set in R^2 with the usual

A: We have to check that boundary of a path connected set in R2 with the usual topology is…

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

A:

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

A: False

Q: Let A be a nonempty subset of R that is bounded below. Define the set B={b: b is a lower bound of…

A: Given A is nonempty subset of ℝ. Also, A is bounded below. Greatest lower bound property: The…

Q: Let S be a set with a finite number of elements, and let f: S - S be a map _ (a) If fis onto, can If…

A: We are given that S is a set having finite number of elements and f : S--&gt;S be a map.Before we go…

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.