5. Suppose that A and B are bounded subsets of R. Prove that AUB is bounded and that sup(AUB) = sup{supA, sup B}.

Q: Show with all the relevant details, albeit succinctly that every compact metric space M is complete.

A: Here we use sequential definition of compactness so that we use that in proving the completeness of…

Q: 2. Let X, Y be nonempty sets of real numbers that are bounded above. Let X + Y = {x + y:x € X and y…

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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}.…

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

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Q: (2) Find all connected covering spaces of RP2 v RP².

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Q: Consider the reglon bounded by the graphs of y- and y- VE

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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…

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Q: 10/Graup Než,2,3,4,---} has no internat Points Trucor Fats then lim foxteok Trucurfak

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

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

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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: Prove (sn) is bounded if and only if lim sup |Sn| < +∞.

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Q: 3.2 Prove that in any metric space (S, d) every closed ball S,ro] is a closed set.

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Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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Q: 3.) Show that the on( giln by norm (an hot be given bu a scalor produd

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Q: 2) Show that the digital line topology on Z is not Hausdorff.

A: Given that Digital line topology on ℤ, we have to prove that it not Hausdorff Therefore, Let a space…

Q: Show that R is not compact.

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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: Suppose metric space X is not path-wise connected then X is not connected. O True O False

A: Question 9 Answer: as a path connected space of employees connected but converse is not true. So I…

Q: of points of S. 6.30 Show that a set consisting of a finite number of points in E is bounded. Gnita…

A: Given, a set consisting of a finite number of points in E.

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

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

Q: Prove that every finite T,- space has the discrete topology.

A: T=(X, τ)  (X, τ) is a T1 space if and only if all points of X are closed in T.   Let X≠∅ be a set.…

Q: Give an example of a closed subset S of R^2 such that conv S is not closed

A: An example of a closed subset S of R2 such that conv S is not closed is given below.

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: If S, S, are two subsets of an inner product space V(F) then show that

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Q: Show that R³ is spanned by S = {(1,1,1), (1, –1, 0), (0, 1, – 1)}.

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Q: umat is bounded, and is not compact in Q.

A: Here the closed means closed with respect to subspace topology. And compact means every cover has…

Q: Show that R³ is spanned by S = {(1,1,1), (1, – 1, 0), (0, 1, – 1)}.

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Q: 3.) Prove that P, is a all ve ctor by showing space 10 0xioms ore

A: proof P2=a0+a1x+a2x2:a0,a1,a2∈R   is vector space over + and ×first we proof P2 is abelian group…

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

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Q: Prove that the Joukowski map 1 p(2) : (z + maps ONTO T - (-1, 1)

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

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Q: Prove by sequences that the open balls in R^n with the usual (Euclidean) metric are not compact

A: In this question, we have to show that open balls in ℝn with the usual metric are not compact by the…

Q: Prove that {4- ) is an open cover of (0, 4), but that this cover has no finite subcover. k=1 This…

A: The complete solutions are given below

Q: If A is bounded from the below, then the set of all lower bound of A has infimum but may or may not…

A: Given:               if A is bounded from the below, then the set of all lower bound of A has:…

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

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

Q: Let S = {(1,1), (1, 2), (2, 1), (3, 2), (3,3)}. Consider the polytope conv(S). Write this set as {r…

A: The given set is S=1,1,1,2,2,1,3,2,3,3. We know that the polytope:…

Q: Consider the mraerespace (R. BIRI ) Lt fir)*3 and ya)= if 1 Then

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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: Show that R3 is spanned by S = {(1,1,1), (1, –1,0), (0, 1, – 1)}

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Q: Suppose that X = {x₁,x₂,.,x} is a subset Inner Product space. Which option is correct? a. b. C. d.…

A: Given: Suppose X=x1, x2, x3,…, xn is a subset Inner Product Space. We have to choose the correct…

Q: Find a polyomial f(x) of degree 3 that has the following zeros. -8, 3, 0

A: Given :  To determine the polynomial fx of degree 3 that has the following zeros : -8, 3, 0

Q: Theorem 7-6. If the variables are uniformly bounded then the condition, lim Bn = 0, n- 00 n2 is…

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Q: Suppose metric space X is not path-wise connected then X is not connected. True False

A: A space is said to be Path connected if we can fine a continuous function between any two points of…