Let (R,d) be diserete metric space then R is not compact . True or false.??

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

A: This is a problem of Topology, Functional Analysis.

Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

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Q: If the dual X' of a normed linear space X is fininte dimensional, then X is finite dimensional

A: This is the question of Functional Analysis.

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

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Q: Let (R. d) be discrete metric space, then Ris separ able O True O False

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

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Q: If X =R and d(x.y)30, then X metric space not metric semi metric

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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

Q: Let (R. d) be discrete metric space then R is not complete O True O Faise

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Q: メ=D¢ c omplex number %3D Doyou CGid) is metric space of not ??

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Q: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

A: T2-space: A topological space X is said to be T2-space for given x, y∈X, there exists open sets U…

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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Q: 2) let X and y be metric spaces and T. XY Y a ContinUOus mapping Then the image of a Compact subset…

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Q: A completely regular space is also T1-space 1 True 2 False

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Q: 2. If X,Y are Banach spaces on XOY define |(x, y) = |, +|yly . Show that this gives a norm making X…

A: Given: Dual of c0  is isometric to l1 State the principle of uniform boundedness.

Q: Ex: Let d is a trivial metric o space ) show that every infir

A: We have given trivial metric space d on infinite set X. We have to show that every infinite subset…

Q: Olet se be u lecally compact topological space. Prove that that GL2) in a cEalyehra (ie. prove (a)…

A: Given:- Ω- Locally compact topological space. We have to prove that C0(Ω) is a C*-algebra means,…

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and ВС (Х, М) :%3D…

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Q: . Prove or disprove every metric space is normed space,

A: A normed vector space or normed space is a vector space over the real or complex numbers, on which a…

Q: If (X.r) and (X,r,) are T,-spaces, then (X.r,nt) is also a T, -space. GOOD LUCK

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Q: a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b.…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

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

A: According to the given information, it is required to show that all the completions of a metric…

Q: rove that spaces Ip for os p<l If for pe(0, 0) {0} and complet. are

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Q: With the help of exomples Show thaat on finite dimensional are equivalent. any two norms Linear…

A: Given: Any two norms on finite dimensional linear space are equivalent.

Q: give example is not To-SPace about to folegical SPace

A: We have to find a topological space which is not t0.

Q: 3) Consider the topology ↑ {GCR: 0 ¢ G or G = R} on R. Is open = %3D interval (-1, 1) compact in (R,…

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Q: Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both…

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Q: If (X.r,) and (X,r,) are T.-spaces, then (X.r,nr,) is also a T,-space. GOOD LUCK

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Q: Let X and Y be compact spaces. Then X × Y is compact.

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Q: let (2, m) be a meesurable space. then 1f f9: x LJare measural ble function, then -> ) f43 is mea…

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Q: IJAis dense in a metric space X. then A is closed True O False O

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Q: Show that the closed ball Y in a metsic SPace (X,d)is a closed set. Also, show that it (x,d) is…

A: Show that the closed ball Y in a metric space (X,d) is a closed set. also show that if (X,d) is…

Q: Prove the following: Let (Y, d) be a metric space. If Y is totally bounded, then Y is bounded.

A: Given:  (Y,d) is a metric space, and Y is totally bounded To Prove Y is bounded Definition: Let…

Q: Consider the spaceZ+with the finite complement topology. Is thisspace a Hausdorff space? Is this…

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Q: Show that the closed ball Y in a metric space (X, d) is a closed set. Also, show that if (X, d) is…

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Q: c spaces. Show that X x Y and Y

A: The sup-metric d1 on X×Y is defined as d1x1, y1, x2, y2=max dx1, x2, ey1, y2. Similarly, the…

Q: (a) Prove that every nonempty proper subset of a connected space has a nonempty boundary. Is the…

A: As per Bartleby guidelines, for more than one question asked, only first is to be answered. Please…

Q: Define what it means for a topological space to be completely regular,

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Q: Let X=R and T = {]a, +oo[/a R}U{o,R}, then (X, 7) is a regular space. True False

A: This is a problem of topology.

Q: ii) True or False: The space R is homeomorphic to the closed interval [-1,1] with the usual…

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Q: (a) Show that every metric space is T2-space.

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Q: Show that the set of all real number constitutes incomplete metric space an we choose doxoy) = Itan'…

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Q: Prove disprove that (R.Tco-finite) is Te-space or

A: To Prove - Prove or Disprove that ℝ, τco-finite  is T2 - space.       Definition used - T2 - space…

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

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Q: tet is cosinte toploy on set R show that (R, t) is O NOVmal space- Regualra space

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Q: a) Find a space that is a regular but not Hausdorff. Justify your claim.

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

Q: For a normed space X, prove that the dual of X is separable implies that X is separable. Is the…

A: NOTE: According to bartleby we have to answer only first question please upload the question…