(b) Beweisen Sie, dass eine Abbildung d: A× A →→→→R genau dann eine Metrik auf der Menge A ist, wenn (4 P.) Vx, yЄA: d(x,y) = 0 ⇒ x=y (1) und Vx, y, z ЄA: d(x, y) ≤ â(z, x) + d(y, z) (2) gelten.

Q: Let R = - b ,be Z}and let ø : R→Z be a mapping defined by : b = a-b. a a

A: We will find out the required value.

Q: (X,e) Let be a metric space The diky-k.glry (KEIR) function defined from XxX to IR is a metric on X,…

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Q: Exercise 6. [Topology] Let X be a nonempty set and 7 = {U C; U = 0 or X\U is countable}. Prove that…

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Q: Let c[o] be with metrics the Space d, (f, g) - max f(x)-gen) fen) = x2 4 gcx) =x The find d, Cf,g)…

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Q: Let (X,d) be metric space and X is unbounded, then d (z. y) . y EX, is 1+d(r,y) di de fined as dy =…

A: Metric space

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Q: Q3/(a) Let d be a trivial metric on a set of X, find the induced topology on X by a metric d. (b)…

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Q: 4) Let T be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (R, 71) →…

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Q: Let (x,d) be metric space and A, BC X. 2. Show that (AnB)= A'n B and AUB = AUB %3D

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Q: (a) Define a metric space (X, p). (b) Let p: R+ → R+ be a function defined by ln (2). Prove that p…

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Q: Which of the listed mappings is an isomorphism of vector spaces?

A: (a) A:R3→R3​[x],A:(a,b,c)↦ax+bx2+cx3 Linearity:…

Q: cise 2 Let X be a topological space. - Show that X itself is a Baire space. T

A: Given: X is a topological space. To Prove: We have to show that the topological space X itself is a…

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Q: Show that: All completions of a metric space are isometric.

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Q: C. Show that a set A in R² is open in the Euclidean metric it is open in the max 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: 2.8.12 Let X be a metric space. Prove that an arbitrary intersection of compact subsets of X is…

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Q: Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only…

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Q: 2.8.15) Prove that a compact metric space is both complete and separable.

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Q: Let X be any nonempty set, and define (o if x = y. 11 if ry. Prove that (X. d) is a metric space.…

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Q: Let f be a continuous mapping of a metric space X into metric space Y,g be a continuous mapping of…

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Q: Consider A = [3,7), B = (7,9] as a subset of R with the usual metric  dR (x,y) = abs(x-y). Show Ac…

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Q: LBe M, N subsets of space W. Then span (MUN) = span (M) + span (N)“ %3D

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Q: Exercise 1. Let X be a metric space. The subsets of X which are arbitrary (finite or infinite)…

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Q: Let (C[a,b] .d) be the discrete metric space. Then Cla,b] is bounded. * False True

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Q: Solve the following equation 7y-4(5y-1)=6-(7y-12)  Solve for n in the following equation…

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Q: Let (R, d) be discrete metric space and A = N then A is complete and compact neither complete nor…

A: Given that (ℝ,d) is a discrete matric space and A=ℕ. then     d(x,y)=0 if x=y1  if x≠y A matric…

Q: Algebra Consider the set B = {(x, y) ∈ R 2 : 0 < x < 1, 0 < y < 2}. Prove that B is open in R 2…

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Q: Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric. There are…

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Q: If A is a subset of a me

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