18 What is the size of the space stretched by the set {x, x+x², x³, 1 , x²} ?    a) 2nd  B) 34  NS) 4  D) one  TO) 0

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Q: Q7: Prove that (R", |l-|| ) is complete metric space.

A: This is a problem of Topology, Functional Analysis.

Q: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

A: Show that the closed ball Y in a metric space (X,d) is a closed set. Also show that if (X,d) is…

Q: Prove that in a metric space (S,d) every closed ball Sr[Xo] is a closed set

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Q: Let (R, d) be usual metric space, then Ris ot separable True False

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Q: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

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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

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Q: Let (X.d) be mnrtric space and X is closed then X is O complete Ono decsion Onot complete

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A: Consider, I=A⊂X : Ac is countable ∪ϕSince,Xc=ϕ is countable, X∈I Also given, ϕ∈I Assume, Aii∈N∈I…

Q: 3.5 Show that the metric space (R, d), where d(r, y) = |x – y|, is separable.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology.   We claim that Z is NOT…

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: In order to be dense in R, R has to be equipped in the Tp. Also as in the interval, 5 is not…

Q: Be M, N subsets of space W. Then span (MUN) = span (M) +span (N)- %3D

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Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If AC B,…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

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Q: give a Reel number r>0 and a∈ X in (X, ||.||) norm space. Determine whether B[a,r] is convex by…

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Q: Q: In usual topological space (R.Tu). answer the following: 1- Is [0,1] is compact set in R? Why?…

A: Note: Hi! Thank you for the question, as per the honor code, we are allowed to answer three…

Q: Show that close ball Y in a metfic space (X,d) is a closed set also show that if (X,d) is complete…

A: Show that closed ball Y in a metric space (X,d) is a closed set also show that if (X,d) is complete…

Q: Prove that: 1. (R, |. |) is a complete metric space. 2. (Q.I-1) is not complete metric space.

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Q: Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact

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Q: Prove the following sets are not compact by finding an open cover that does not have a finite…

A: a) Given-  A set A=−1,1. Explanation- Consider a set Xn=-2,1-1n for n=1,2.... Then clearly,…

Q: If S be a n-dimen Sional normed Space, then prove that its dual space is al so n- dimen Sional .

A: A normed space E is called a Banach space if it is complete, i.e., if every Cauchy sequence is…

Q: LBe M, N subsets of space W. Then span (MUN) = span (M) + span (N)“ %3D

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Q: In the Triangle Inequality of Euclidean n-space: Prove ||x - z || <= ||x - y|| + ||y - z ||…

A: The triangle inequality, in Euclidean geometry, theorem that the sum of any two sides of a triangle…

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. biso (2) If…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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Q: Show that the closed ball Y in a metsic SPace (X,d)is a closed set. Also, show that it (x,d) is…

A: Show that the closed ball Y in a metric space (X,d) is a closed set. also show that if (X,d) is…

Q: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

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A: A metric space X is compact if every open cover of X has a finite subcover.

Q: Q7: Prove that ( IR", ||·|| ) is complete metric space.

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Q: 1) Prove that for any space, X, with any topology, T, that: (i) CI(X) = X and (ii) Int(X) = X.

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Q: Prove the following: Let (Y, d) be a metric space. If Y is totally bounded, then Y is bounded.

A: Given:  (Y,d) is a metric space, and Y is totally bounded To Prove Y is bounded Definition: Let…

Q: 3) Consider the topology 7 = {G CR: 0 ¢ G or G = R} on R. Is open T 3= interval (-1,1) compact in…

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Q: Show that the closed ball Y in a metric space (x,d) is a closed set .Also show that if (x,d) is…

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Q: Show that the closed ball Y in a metric space (X, d) is a closed set. Also, show that if (X, d) is…

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Q: Q7: Prove that (R", ||-|| ) is complete metric space.

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Q: 4) Consider three-dimensional Euclidcan space R with the usual Euclidean distance, definition of…

A: This is a problem of non-euclidean geometry, euclidian postulate.

Q: Amongst the various possible products of four topological spaces A, B, C and D, show that (A × (B ×…

A: We shall solve this in next step

Q: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.

Q: Assume X is Hausdorff. If K C X is compact and x ¢ K, show that there exist disjoint open sets U and…

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Q: Prove that in a metric space (X, d) every closed ball that is a set K(x, r) = {y e X : d(x, y) <=…

A: Question: Prove that in a metric space X, d every closed ball K¯x, r=y∈X, dx, y≤r is a closed set.…

Q: Prove that if X,d is a metric space and G ⊆ H ⊆ X, then G-bar ⊆ H-bar.

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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,…

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Q: ii). what { disie [4NI} to fobrm vedtor space ? set vectas a bauis for en N-dimeneional alte the…

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Q: Consider the following subsets of R equipped with the Euclidean topology: Зп — 1 A = {- n+1 B = Ü 1+…

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Q: Prove the following: Let (Y, d) be a metric space. Let B be a bounded subset of Y. If A C B, then A…

A: Let Y, d be a metric space and B be a bounded subset of Y. Suppose A⊂B. To prove that A is bounded…

Q: 2) Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

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Q: (a) Prove that the metric space (Z,) is complete.

A: Complete Metric Spaces: A metric space (X,d) is said to be complete iff every Cauchy sequence in X…