Q7: Prove that (R", ||-|| ) is complete metric space.

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A: 3.3 is also known as Hausdorff Property. d(x1, x2) > 0 for every x1 ≠ x2   (metric space…

Q: c. Suppose that G,H < Sym(Q). Show that the corresponding spaces (2,G) and (Q,H) are equivalent if…

A: Suppose that G, H ≤ Sym (Ω). Let G, H be a multiplicative group and let Ω be a set. An action of G,…

Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

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Q: If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is l.

A: We will prove the given statement.

Q: Prove that L∞[a, b] is complete with respect to the norm || · ||∞.

A: The elements of L∞[a,b] are the essentially bounded measurable functions.

Q: Let E1, E2, ..., En ⊂ X,d be compact subsets of a metric space X. Just using the definition of…

A: For a metric space we know that the for a compact set every open cover has a finite subcover. So…

Q: (b) Prove that the set S = {x EQ: √2<x<√3} is closed and bounded in (Q.1-1), but not compact in…

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Q: Provethat amappi'ng from a metric SpaceXto metri'c space X is con tinuok ifand only f the inverse…

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Q: Show that every plane through the origin in R³ may be identified with the null space of an element…

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Q: Q7. Assume that (R,T) is the usual Euclidean topological space. N = {1,2, 3, ....} C R. Determine…

A: There is no interior point of N.

Q: 3.5 Show that the metric space (R, d), where d(r, y) = |x – y|, is separable.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: (a). Define three different metrices d1, d2, d3 and their induced norms on R³. Also prove the…

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Q: Let (X, d) be a separable metric space. into l∞.

A: This is a problem of functional analysis.

Q: Consider IR as a metric space.with metric dux, y) = 1x-y1. Show that the open interval C0, 1) is an…

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Q: If M is a closed linear subs r space X, and x, be a vect the distance from x, to M, ional f, in X'…

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Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

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Q: Let ( Xr d be ame tric space,show that exist atopology on X induce b

A: Let (X, d) be a Metric Space We know that in a Metric Space  (1) arbitrary union of open sets is…

Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is l.

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Q: Q2: In Euclidean metric space (R², 1. 1), is the set A = {(x, y): xy < 1} compact? giving the…

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Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: Q3 Assume that (M,d) is a Compact metric Space. Show that, f: (M,d) → (Y, 2) is Continous bjective…

A: f:(X, d) to (Y, d) is said to be homeomorphism if  1) f is continuous.  2) f is bijective.  3) f…

Q: A. Let H be the set of all points (x, y) in R? such that x2 + 3y2 = 12. Show that H is a closed…

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Q: Assume that (M, d) is a compact metric space. Show that if f: (M,d) → (Y, d) is continuous and…

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Q: let X be a firite dimensional notm Space Then Prove that is compat.

A: As X is a finite dimensional norm space, this means that, dimX is finite.

Q: Q # 3(b) Let (G, H) be a disconnection of a topological space (X, 7), ant let AC X. If A is…

A: Before going to this proof, we prove the below statement which in turn proves the above result.…

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A:

Q: (b) Show that the metric space (Q,1) is not complete. (c) Prove that the discrete metric space (M,d)…

A: Complete Metric Spaces :- A metric space (X, d) is said to be complete iff every Cauchy sequence in…

Q: If A is 3 × 3 with rank A = 2, show that the dimension of the null space of A is 1.

A: Given: A is 3*3 matrix with rank A = 2

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

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Q: Let K be a compact subset of a metric space X, and assume that -R is continuous. Prove that f…

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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…

Q: 1. Let M, d be a metric space with A C B C M. Suppose that B is totally bounded. Prove that A is…

A: Let (M, d) be a metric space and let A be a nonempty subset of M. The diameter of A is defined as…

Q: Let x be an interior point of a subset S of a metric space (X,d). Show that x must also be a limit…

A: Interior Point: Let A be a subset of metric space X, d. A point x∈A is called an interior point of A…

Q: let a-Rbe ametric sPacesConsir he following subsed ofP @ Show that Dra De are comPact Setsornot of D…

A: Question from real analysis.

Q: (d) Consider the set [1, √5] C [−1, √10]. Explain f is not continuous by connectedness, compactness,…

A: Given: The function, f:-1,10→ℝ defined by, fx=x2,x&gt;2x,x≤2 The set 1,5⊂-1,10. To explain: Why the…

Q: Q2: In Euclidean metric space (R2, 1. 1), is the set A = {(x, y): 2x² + y² = 1} compact? giving the…

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Q: Use the Completion Theorem below to show that the Completion of the discrete metric space X is…

A: By completion theorem, there exists a complete metric space which has a subspace that is isometric…

Q: Suppose U and v are conformally equivalent. Prove that if U is simply connected, then so is V.…

A: Given that U and V are conformally equivalent that is there exist a mapping f from U to V such that…

Q: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

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Q: Q7. Assume that (R, T) is the usual Euclidean topological space. N = {1,2,3,..}c R. Determine N°,…

A: ANSWER :

Q: 1. Show that if & is a function from a nonempty compact metric space X to itself such that d(@(x),…

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Q: H.W.2 Exercise Prove that: 1. (IR, J- |) is a complete metric space. 2. (Q. |-1) is not complete…

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Q: Q6: (a) In any metric space (S, d), prove that every compact subset T is complete. (b) Prove that…

A: A metric space is a set together with a metric on the set. The metric is a function that defines a…

Q: c) Let (M, d) be a metric space and let S,T≤ M. Prove that SOT = ST. But SOT SOT.

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Q: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

Q: 5. State whether the following is true or false m) A subset I of a metric space R with the usual…

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Q: Show the following result: Theorem. Let (X, d) be a metric space, A C X, and let u: A → R be an…

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Q: Let x1 and x2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

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Q: Prove that: 1. (R, J. |) is a complete metric space. 2. (Q, I. |) is not complete metric

A: To prove that R,. is a complete metric space.  Suppose that xn is increasing and bounded, Let…