4.1 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set. 4.2 Let F be a closed set for k = 1, 2,..., n in (S, d). Show that Fr is closed. k=1

Q: C^k is the set of all functions that are continuous and have first k derivatives that are…

A: Ck=set of all functions that are continuous and have first k derivatives that are continuous To show…

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

Q: et [a, b] be a real interva b]→X is continuous, the is continuously differentia

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Q: Suppose the function g is on the interval [a, b], and |g(x)|< B for all x E R. I) Prove that |g(x)?…

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

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

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

Q: 3. Let (X, d) be a metric space and x1, x2, ... , Xn, ... a sequenc

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Q: 3. Let X and Y be normed spaces and f:X→Y be a bijection linear function. prove or disprove f is…

A: It is true that if f is a bijection linear function from a normed space X to a normed space Y, then…

Q: Let f be a continuous real-valued function defined on a metric space (X, d), and let D be a compact…

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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Q: 2.4.10. Let X be a metric space. (a) Suppose that X is complete and F1 2 F2 2 ··. is a nested…

A: Let X be a metric space. (a) Suppose that X is complete and F1⊇F2⊇⋯ is a nested decreasing sequence…

Q: 11. Let (X, d) be a pseudometric space, and suppose d has the property that d(a, b) >0 whenever ab.…

A: 11) To prove that every finite subset of X is closed, we need to show that its complement is open.…

Q: 10. Let (X, 71) be a Lindelöf space and (Y, T2) be a compact space. Then show that the carte- sian…

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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Q: Prove: Let X and Y be two topological spaces. Then the function f:X →Y is continuous if and only if…

A: This is a problem of topology.

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: Q. No. 2 Let N be the set of natural numbers and N = N U {∞}. Define a function d: N* × N* → R as…

A: I have proved the three conditions for metric

Q: Let (X,7) be a topological space and ACX. Show that (n) A is open if and only if Int(A) = A. (b) A…

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

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: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and ВС (Х, М) :%3D…

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Q: 2.1 Let B be the set of all bounded sequences of real numbers and define the function d : B × B → R…

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Q: Q7) (a) Let X be a Banach space and R the usual normed space. Show that ||(z, a)|| = ||||+|al is a…

A: (a) Given a Banach space  and  the usual normed space, we want to show that the norm  is complete on…

Q: 1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B → R by…

A: we have to solve given problem:

Q: Let (X,d) be any metric space, then the only open sets and closed sets are and X O True O False

A: Given statement is false because if we take descrete metric space

Q: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

A: Given: A subset (a, b)⊆ℝ is T2-space in ℝ, Tu. To determine: Whether given statement is true or…

Q: Q2:- Let (X, d) be a metric space. Xo, Yo EX and r, s > 0. Show that B, (yo) Br(xo) if s+d(xo, yo)…

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Q: 4. Let A C R" be both connected and sequentially compact. Prove that if F: A –→ R is continuous,…

A: Consider the given information.

Q: 2.1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B R by…

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Q: (b) Let X = {0, 1} and let (X*,d) be metric space, where d(x, y) = the number of elements of the set…

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Q: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: 32. Show that, in a normed space, x x implies -(x1 ++Xn) → x.

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Q: 11. Prove that the function d: Cx CR defined by d(x, y) = Solle is a metric on the set of all…

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Q: 5. In any metric space (M, D), prove that D(a1, an) ≤ D(a₁, a2) + D(a2, a3) + + D(an-1, an), for a1,…

A: Given M,D is a metric space.   The triangle inequality holds in a metric space, which means for…

Q: IfS is a subset of an innerproduct space V(F) then show that Scs=(0}. %3D

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Q: (b) Show that xE R, x2 - 4x + 5 > 0.

A: As this is a multiple question type, only first question is to be solved, according to the Bartleby…

Q: We note that ℤ2 which means ℤ*ℤ. Let d: ℤ2 *ℤ2→ℝ≥0 be the function that is given by…

A: The given in this question is a function d that takes two points (x0,y0) and (x1,y1) in the set of…

Q: Let F and G be nonempty bounded sets. Prove that inf (F UG) = min{inf F, inf G} and sup (F UG) =…

A: Since F and G are non-empty bounded subsets of R, sup F, sup G, inf F and inf G are all exist in R.…

Q: 5. Assume that f:Z+→Z*. Then the type of the function f(x) = LxJ +2 is considered a) Onto function…

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Q: 6. A metric space (M, D) is called ultra-mteric if D(a, c) < max {D(a, b), D(b, c)} for all a, b, c,…

A: In this problem firstly I have shown that (M,D) is a metric space.secondly I have shown that D is…

Q: R and C with ||x|| = |x| are Banach spaces.

A: Given: ℝ and ℂ with x=x.  We have to prove that they are Banach spaces.

Q: 5. Let (X, d) be a compact metric space and {K) a decreasing sequence of closed sets in X. If : XX…

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Q: 2.53 Let B[a, b] denote the set of all bounded functions on [a, b). Is Bla, b] a vector space? Prove…

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Q: Q5. Let (X, B(R), µ) be a Borel measure space. Suppose that {A, = ["+1_ 4n-1 Show that A1 C A2 C…

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.