4. Let (X, d) be a metric space and d' be a metric on X that is equivalent to d. Let and be the topology on X induced by d and d' respectively. Show that 7 = I'.

Q: 1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {phi_t}) be  a flow on a…

A: The objective of the question is to define a flow on a metric space, identify a fixed point of the…

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, d) be a metric space, prove that for any x, y, z E X, we have |d(y, z) - d(x, z)| ≤d(x, y).

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Q: ,A, B,C be 4 points in R², no three of which colinear. Fix A' e VA, B' e VB and C' e VC h that AB ||…

A: In the following question, concept of parallel line is applied. Parallel Line Parallel lines are…

Q: 3. This question concerns the L metric d and the L¹ metric e on a) Calculate d(f.g) and e(f.g),…

A: Consider L∞ metric d and L1metric e on C[0,1]. (i) To Find: d(f,g) , e(f,g) for  f(x)=1+x , g(x)=x2.…

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: For each case, determine whether the metric space is complete or not. Support your answers. (i) Z…

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Q: let AABC be a triangle S². Let the dual point C'ES² of C be defined by the following three…

A: Given: △ABC is a triangle S2 (the two-dimensional sphere). The dual point C'∈S2 of C is described by…

Q: Suppose (X, d) is a metric space and let r> 0. Show that p: X2 → [0, c0) defined by p(x, y) = r·…

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Q: d, d': xxx-R be two metrices on x that Satisfy following for every nEX and >0 7>0 st. Bai (x, S) C…

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Q: 2. Prove that if d is a metric for X, then another metric d' for X is given by if a ≤ b " if b ≤ a…

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Q: 1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {ϕt}) be a flow on a metric…

A: The objective of the question is to define a flow on a metric space, identify when a point is a…

Q: 3. Let d₂(x, y) = |arctan(x) — arctan(y)| for x, y € R. (a) Show that d₂ is a metric on R. (b) Are…

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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: Let d be a trivial metric on a non-empty set X and d₁: XXX → R such that x = y d₁(x,y) = { 2 if x…

A: Given:d1:X×X→R such that  d1x, y=0if x=y2if x≠y∀x , y∈X To Show: d and d1 are equivalent.

Q: 6. Let S = R — {0} = (-∞, 0) U (0, ∞) with the usual metric. Show that (-∞, 0) is a closed subset of…

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

A: Let (X,d) be a metric space, where d is the discrete metric. Then, for any x,y∈X, we have: d(x,y) =…