Given the bow-tie T:= {(x, y) ER: [y] < 1, y ≥ |x|} (a) Sketch the set T, and determine its boundary T. (b) Is the set T compact? Justify your answer. (c) Prove that the intersection of a finite number of compact subsets of a metric space is compact.

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

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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, 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: (b) Is (R2,d) a metric space, where d((x.y1).(x2.y2)) 1x1-x2l; for all (x, , y1 ), (x2 2) E R2 ?…

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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: (f) The set B = {(z, y) € R? : ² + y? < 1} as a subset of R? with indiscrete metric is disconnected.…

A: As per guidelines we have to attempt 1st three parts

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

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Q: (3) Let R² and R be endowed with their Euclidean metrics. (a) Show that projections p₁(x, y) = x and…

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Q: For Exercises 1-6, state the 1. f(x, y) = x² + y²-1

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Q: Material

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Q: let Qid) be a metric space and AiB Bubeet of x ACB = d(A) cd(B) [ Soi d(A) is closed set] P B is…

A: Let X,d be a metric space and A, B⊆X.   We will use the following in the proofs, Definition of clA:…

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Q: (7) Which of the following is disconnected space: (a) (R\ {0}, Tusuat) (b) (R, Tusuat) (c) (X, t).…

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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…

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Q: Q7: Prove that ( IR", ||·|| ) is complete metric space.

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Q: Let (R, d) be discrete metric space, then R is not separ able O True O False

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Q: (a) Prove that every closed subset S of a compact metric space (M,d) is compact in M.

A: Note: Our guidelines we are supposed to answer only one question. Kindly repost other question as…

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Q: 5.1: The following stat- X is disconnected.

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Q: Define a metric as d(x,y)=|x1-y1|+|x2-y2|. Find the set of all points x=(x1,x2) in R2 such that…

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A: Given A⊂X where X is non-empty and A≠X Suppose, T=X,∅,A,B be a topology.  Here B=A∪frA To prove for…