d(r, y) Let (X, d) be a metric space and let p(r, y) = for all r, y E X. Show that (X, d) 1+d(x, y) is complete if and only if (X, p) is complete.

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: Draw the Taxi-metric hyperboloid |x| + |y| + |z| = 1

A: As per the question we are given the equation of Taxi-metric hyperboloid as : |x| + |y| + |z| = 1…

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: Verify that {u₁,4₂} is an orthogonal set, and then find the orthogonal projection of y onto…

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Q: Let X = (1, 2, 3), Tx = (0, X, (1), (2, 3}}, then (X, Tx) is %3D %3D 1) a T1 space 2) a T2 space

A: T1 space: A topological space X has the T1 property if x and y are distinct points of X, there…

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: f) Every continuous map T² RP2 is homotopic to the constant map.

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Q: Show that the line [x, y, z]= [5,-1,3]+[4,-5,2] lies on the planes 3x+2y-z = 10 and 7x+4y-4z=19.

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Q: Define p: R^2 x R^2 to [0, infinity) by the rule p(x1,y1),(x2,y2)=squareroot((x1-x2)^2+(y1-y2)^2)…

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Q: X Edit 10. Rectangle GHU wilth verlices G(2.-1), H7.-1), 17.-4), and

A: Graph plotting

Q: Calculate Non-zero Christoffel symbol for the metric ds² = −dt² + A² (t)dx² + B² (t)dy² + C² (t)dz²

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Q: Show that set : A = {(x,y) = R² : sin(x) ≤ y ≤ 2e-2²] is closed on Euclidean plane R².

A: The given set is: A=x,y∈ℝ2: sinx≤y≤2e-x2. We need to shoe that the set A is closed in Euclidean…

Q: Calculate Non-zero Christoffel symbol for the metric ds² = -dt² + A² (t)dx² + B² (t)dy² + C² (t)dz²

A: As per the question we are given the following 4D metric :ds2 = -dt2 + A2(t)dx2 + B2(t)dy2 +…

Q: Give an example of (v1, v2, v3) and (w1, w2, w3) independent se- quences in the R“ such that…

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Q: Give an example of (v1, v2, v3) and (w1,w2, w3) independent se- quences in the Rº such that…

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Q: real line ℝ and complex plane ℂ  are complete metric space

A: Definition  Complete Metric Space : A metric space (X,d) is called complete if every Cauchy Sequence…

Q: Determine whether u, v and w lie in the same plane when positioned so that their initial points…

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Q: (b) Every continuous map S? →T² is homotopic to the constant map.

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Q: What is Euclidean Space??

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Q: 2. Do the three planes X + 2y + z = 4, y - z = 1, x+ 3y = 0 have at least one commom point of…

A: The equation of the plane is given by the formula ax+by+cz+d=0. The system of planes can be solved…

Q: (d) Passing through the point (0, 1, 2), perpendicular to the line L and intersects L. L(xo. v) =…

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Q: Sketch the domain D in the (x, y)-plane satisfying ≤ y ≤ 2e- and e ≤ y ≤ 3e. In your sketch, label D…

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Q: (4) 2 What is the square of the (minimum) distance between W and P? Let W Span< = -2 be a plane in…

A: Given that

Q: Find the distance from y to the plane in R³ spanned by the set (U₁, U₂} 6 - 2 4 y = -9 a. b. C. 43₁…

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Q: Show that (Cod) is metric space

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Q: - {z,y, 2)| 2 2= > 0} //| dpdød0 set Consider the 3D domain W = Calculate | a2 + y? + z² d(x, y, z)…

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