Question 4: Suppose that the metric d on M is the discrete metric. (a) Show that every map f: M →Y is continuous. (b) Show that if M = R", then the map Id: (R", d) → R" : 1+I is uniformly continuous. (recall that when no metric is specified on R", its standard metric is assumed. Also, d is clearly the discrete metric.)

Q: (a) Prove that f is a one-to-one correspondence. (b) Prove that ray AF is between rays AB and AE if…

A: (a) To prove that f is a one-to-one correspondence, we need to show that f is both injective and…

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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: Let (R, d) be usual metric space, then Ris ot separable True False

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Q: Let C be the closed curve indicated in the diagram. Use Green's Theorem (2,1) (o,i) (x² + y?, y³ +…

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Q: where a, b e X, is also a metric on X. 2- Let (X, d) be a metric space and T the corresponding…

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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…

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Q: Consider the patch o (u, v) = (u, v, u² + v²) here u, v E R. Find the formula for the Gauss map G,…

A: Part(a): Given function is, σu,v=u,v,u2+v2      u,v∈ℝDiffrentiating partially with respect to…

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

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

Q: 3: Let a function d: R* x R* → R defined by d(x, y) = E-\xi - yıl Vx = (x1,...,x4), y = (y1,...,y4)…

A: Here we will prove that d is a metric function.

Q: Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed…

A: Given : As per the question we are given the following bivariate function : x2 + xy + 3y2 = 3 And a…

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Q: Decide which of the following are metrics on X = R2. For each, we let x = (x1, x2) and y = (y1, y2)…

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Q: 3. Consider the circle as a vector-valued function r(t) = cos(t)i + sin(t)j. Calculate limt→+∞ r(t)…

A: Find the limit

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: Consider the following mappings on RXR: * d1(x, y) = e2z-y, d2 = |2x – 2y|, and d3 = VI2x – y|. %3D…

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Q: Consider the following mappings on RXR: * d1(1, y) = e2"-y, d2 = |2x – 2y|, d3 = V[2x - yl- and |…

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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…

Q: 1. Match each vector function with the graph of the corresponding space curve. Arrows indicate…

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Q: Consider the set E in R2 defined by E= {(x, y) = R²: y = 0, a< x < b}, where a<b. Describe the…

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Q: . Let X be a nonempty set with metric d. Show that the function e defined by e(x, y) = min[1, d(x,…

A: We know that d(x,y) is a metric on X. Hence d satisfies all the properties of a metric.  Now e(x,y)…

Q: Let a = (a, az); b = (b, ,b2) € R?, define the function %3D %3D d: R? x R? ---> R by d=(a, b) =…

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Q: If T is surjective and V = span(3), prove that W = span(T(3)). Hint: given that span(T(3)) ≤ W, it…

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Q: Show that the curve In (2X-1) , n> i has no stationary point

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Q: Let S' denote the set of (x, y) E R × R such that x? + y² This space is called the circle. Define a…

A: S1={(x,y)∈ℝ2/x2+y2=1} and p:ℝ→S1 by pt=cos2πt,sin2πt. To show: p is continuous, open,closed.

Q: is that The necessary and sufficient condition that a curve is a plane cury 4 r 00 T 004 r = 0.

A: