Exercise 4. Suppose (Y, d) is a complete metric space. Consider XC Y as a metric space (X,d) with the induced metric from the metric d on Y. If (X,d) is a completion of X, show that there is an isometry I: X → Y which is the identity on X. What is the image I(X)?

Q: 1. Identify each of the following subsets of the Euclidean plane R2 as open set or closed set or…

A: Explanation of the answer is as follows. As per the rules I am doing first three sub parts in…

Q: 5. Give an example of a curve C C xy plane so that its curvature as a space curve is different from…

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Q: Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric). i. Explain why…

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Q: Let X be an infinite set and let d be the discrete metric on X: {: 0if p=q 1 if p + q. d(p, q) = (a)…

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Q: Let X be an infinite set and let d be the discrete metric on X: d(p, q) = 0if p=q 1 if p+q. (a) Find…

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Q: Given V = {(x, y, z): ax + by + cz = 0}; define: a) The sum and the product so that the given set is…

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Q: Exercise 1. Let (X,dx) be a compact metric space. (a) Verify that dy(f, g) = E 2-"dx(f(n), g(n)) nƐN…

A: let X,dX be a compact metric space dYf,g=∑n∈ℕ2-ndXfn,gn defines a metric on Y=f:ℕ→X to show that Y…

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A: The vacuum Einstein equations are given by:Rμν−12gμνR=0 where Rμν ​ is the Ricci tensor, R is the…

Q: The taxicab metric p on the plane R² is defined by the rule: p((x1, y1), (x2, Y2)) = |x1 – x2| + |y1…

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Q: Let (X, d) be a metric space such that X is a finite set. Show that d is strongly equivalent to the…

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Q: Let X = R, und d be the discrete metric, und A = (3,5) Show that if A is compact or not

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Q: Exercise 4. Suppose (Y,d) is a complete metric space. Consider XC Y as a metric space (X,d) with the…

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Q: b) Let X be an inner product space and x, y e x.Prove that xly if and only if kx + y = |lk | + |y*k…

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Q: What is the dimension of the vector space ab c S= { EM2x (R): a =b=c} de f

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Q: Consider IR as a metric space.with metric dux, y) = 1x-y1. Show that the open interval C0, 1) is an…

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Q: Do the following task. a)  State formally what do we mean by being a metric space of X with metric…

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Q: (a) Give an example of a metric on R2. (b) Is the set of all points (x, y) in R² such that y = x²…

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Q: Let (X, d) be a metric space. Define f: Xx X → R by f(x, y) = = that f is a metric on X. d(x, y) 1+…

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Q: Let C be the space curve represented by represented by & (+) = [(4-2+, 36-2, 61) a.) Reparametrize R…

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Q: Let (X,d) be a metric space. Prove that the function d' :X x X →R given by d (x, y)= min {1,d(x, y)}…

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Q: QI/ a) Suppose that f: [0, co) y is a metric space. Show that d(x, y) = fCo(x, y)) is a metric on X.…

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Q: Consider the sets N = R? \ {(0, y) E R² | |y| > 0}, L = {(x, y) E R² | |x| = |y|}, and let dr,…

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Q: 2. Let (X, d) be a metric space and y E X. (i) Prove that the closed ball B[y,2] is a closed set in…

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Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

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Q: Let (x, g) be a metric space. Is the function d (x, y) = k.g (x, y) (k € R) defined from XxX to R a…

A: Let X,g be a metric space . Given that function X×X to R defined as  dx,y=k·gx,y where k∈R A metric…

Q: Show that an isometry of a plane is completely determined by theimage of three noncollinear points.

A: To Show:An isometry of a plane is completely determined by the image of three non-collinear points.

Q: Consider a surface with a metric tensor 2uv gij = u² 2uv 0²2 where x¹ = u and x² = v. What is the…

A: Given information:Metric tensor , where  and .To find:The connection coefficient .Concept…

Q: Exercise. Prove that in any metric space (X, p), a closed ball {r E X : p(a, x) <r} is closed.

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Q: d(x,y) 1 - Y -

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Q: Is the set S = [0,1] with the discrete metric d separable? Explain

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

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

Q: that det S E

A: Prove that ff is an isometry as,

Q: the metric space symmetry condition of the discrete metric, I find myself writing that if d(x,y)=0…

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Q: 9. verify that lx1l= max ilt)| defines. telasb] a on the Space Ccla, b]. morm Solution :-

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Q: Prove that the d1 (“Manhattan”) metric on R2 is really a metric

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Q: Consider two dimensional hyperbolic space with metric ds² dx²+dy² y² and y > 0. The components of…

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Q: (X,e) Let be a metric space The dixy = k.glny (KEIR) function defined from X x X to IR is a metric…

A: Here Let X,ρ be a metric space. The dx,y=kρx,y, ∀k∈R function defined from X×X to IR is a metric on…