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: (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: 16. A topological space (X, t) is Lindelöf if every open cover of X has a finite or countably…

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

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Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = Ā♂ (A) = A - int A (ii)…

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

Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = A n (A)° = A int A (ii)…

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

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: or not. (c) Let f be a map from a space(X, ). into a space (Y,T) and t,T such that TSt and TS,, show…

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Q: 3.1 Find all limit points of a set A = {(2+ ,2 -n E N} on the Euclidean plane with the usual metric.…

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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: Problem 2. Determine whether the set {: € (2,10] Qy € (1, 2)} is bounded above (resp. below). In…

A: Given set is s=xx-y;x∈2,10∩ℚ,y∈1,2 . We have to determine that s is bounded or not .

Q: Show Corollary 1.24. Namely, show that a measure space (X, M , μ) is complete if (X, M , μ) is…

A: Let X be a measure space with σ-algebra M and the outer measure μ. To prove that X is complete. That…

Q: Let X & Y be normed spaces & x # 204. Then BL (X₂Y) is a Banach space in the operator norm if and…

A: Here BL(X,Y) (or, B(X,Y)), is the space of bounded linear operators from X to Y. First ,we show that…

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: 6 Prove that a closed set in the metric space (S, d) cither is nowhere dense in S or clse contains…

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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: (a) Let || • ||, and || • ||, be norms on a linear space X such that for some r > 0 || x |l, <r || x…

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Q: Determine whether the relation R on the set Z is reflexive, symmetric, antisym- metric, and/or…

A: Given: The relation ℝ on the set ℤ and x,y∈ℝWe need to check whether the given ℝ on the set ℤ is…

Q: Show that Rsta is Hausdorff.

A: Note that a topological space X,I is said to be Hausdorff if (and only if) for every distinct points…

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: Question 3. If X and Y are isometric and X is complete, show that Y is complete.

A: Given below the detailed solution

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: 4. Assume that S, T C R are two non-empty bounded sets. Prove that: (i) SUT is a bounded set; (ii)…

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Q: 5. (8 points) Let S4be the set of all strings of 0's and 1's of length 4. Define F:S4 → Znonneg such…

A: The objective of the question is to understand the properties of the function F defined on the set…

Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = A O (A)° = A – int A %3D…

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

Q: = (4) Suppose that A is nonempty set that is bounded from above and let B Prove that B is bounded…

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

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Q: If A is a subset of a metric space X, then show that A = (x: each neighbourhood of x intersects A}

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Q: 12. Suppose that a, b E R and a < b. Show that the vector space C[a, b] of all continuous complex…

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Q: On the measurable space (R, B (R)), let u: B (R)→ R be defined as: μ(A) = The number of all rational…

A: Given: We are given a measurable space , where represents the collection of Borel sets on the real…

Q: Q2: (a) In Euclidean metric space (R, 1. 1),if A = {y € R: y = 2 cos (2x); x € R} then fined the…

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Q: Find all limit points of a set A = {(2+,2 – 1)|n e N} on the Euclidean plane with the usual metric.…

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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: Exercise 3. 1) Let (X, d) be a metric space. Show that f X → R is Lipschitz with Lipsch constant L…

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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: OR SECTION 13.6 1. Let S be a nonempty, closed, convex set in R" that does not contain the origin.…

A: Consider the provided question,   Suppose, S be a non empty, closed, convex set in  that doesn't…

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: 2. State and prove the Cantor's Intersection Theorem.

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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.