Suppose h: X –→ Y is a homeomorphism and suppose d is a metric on X so that X is complete with respect to d. Recall from Module 7 Homework #5 that p: Y? → [0, co) defined by the rule: p(y1,y2) = d(h-(y1), h-1(y2)) is a metric on Y. a) Show that Y is complete with respect to p. b) Conclude that the property of having a compatible metric which forms a complete space is a topological property.

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: 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…

Q: suppose gradient f(a,b) =4i + 3j. find a unit vector u such that a) Du f(a,b)=0 b) Du f(a,b) is a…

A: Please see the below picture for detailed solution.

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: Find the domain of vector r(t) and find the limit as t goes to 0 of vector r(t)

A: According to question,Given vector,rt = 2-t, et-1t, lnt+1

Q: Let X=R2 and let d be the Euclidean metric.  Define d3:  R2 x R2 to R by d3(x, y) = min{1, d(x, y)}…

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Q: Let X = (0,00). -> a) Show that X is not complete with the standard metric d: X2 → [0, 0o) defined…

A: complete vector space-A metric space (X, d) is  called complete if  the following  condition are…

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: Suppose that we define a function d that maps R2 to R by d(x,y) = { 0,. if x = y 1, if x is not…

A: Given. Let d be a function with the mapping d: R2 → R defined by, dx,y=0,  if x=y1,  if x≠y   (a) To…

Q: Let (t) = (t, t, tº, tª) for all t € R. Define C = Þ(R). Let p = (1, 1, 1, 1). Derive the tangent…

A: Let  for all Let the point be We need to calculate the tangent and normal spaces to C at p.

Q: Let (R>0, d) be the metric space defined by d(x, y) =|log (y/x)|. This metric space is isometric to…

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Q: p(x) = Describe the fibers of y.

A: This is the question of function and mapping. Fibers are actually pre-image under a map.

Q: Let fi(x) E Fun (R,R) be the constant function I, wheret ficx)=1 for all xG RRe Prove that S { sec?…

A: Given: fx∈funℝ, ℝ is a constant function. such that: fx=1……1 S=sec2x, tan2x, fx

Q: For X = (x₁,...,xn), Y = (y₁, ..., Yn) € Rn, define n d(X,Y) := Σ |xi - vi| i=1 (a). Show that d is…

A: Given that, are two vectors.The defined function is: To check the d is matrix space on , have to…

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.