In a Cofinite topology of X the collection of subsets is: a) r = {AcX|A±Ø or A is finite} b) r = {Ac X| A = Ø or A is finite} c) T = {AC X| A =Ø and A° is finite} d) z = {Ac X | A = Ø or A° is finite %3D %3D

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

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Q: For any subset A of a topological space X, we define the boundary of A to be: JA = AnX\A. a) Show…

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Q: We define the included point topology by Tp={ UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in Rif…

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Q: 1. For a subset YC (X,T ) , show that the collection TY ={Y NUJU ET} is closed under finite…

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A: We have to determine the following statements are true or false : i) Let X be an infinite set…

Q: * Let R be with the indiscrete topology. If A = {1,3, 5, 7, . }, then A' = A O N O RO

A: This question is related to topology, we will use the definition of indiscrete topology to solve it.

Q: We define the included point topology by Tp={UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R if

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Q: 3- Let X = {a, b} write a large Topology on X. 4- If A CX, find A' for (X.I).

A: (3) It is given that,  X = a, b. Then we have to write a large Topology on X. (4) It is given that,…

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A: a set A is dense in X means A closure =X

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Q: 1) In a metric space (M, d), show that both Ø and M are open sets in M. 2) In a metric space (M, d),…

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Q: We define the included point topology by Tp={ UcR;U=ø or peU}. Let A = [3,5[, then A is dense in R…

A: In order to be dense in R, R has to be equipped in the Tp. Also as in the interval, 5 is not…

Q: Let T cofinite topology on R and Y={5,6,7} find Ty?

A: We have to construct subspace topology with respect to the cofinite topology

Q: If S is a closed bounded subset of a metric space X, then S is compact.

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Q: For any subset A of a topological space X, we define the boundary of A to be aA = AnX\A. a) Show…

A: (a)

Q: We define the included point topology by Tp={ UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R…

A: R is equipped with Tp and p=3

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If AC B,…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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A: The given theorem is related with the topic product topology. Given that S = π1-1U | U is open in X…

Q: We define the included point topology by Tp={UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R if…

A: The detailed explanation is given below:

Q: 1) In a metric space (M, d), show that both 0 and M are open sets in M. 2) In a metric space (M, d),…

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Q: Exercise 5. 1) Show that the metrics dp, p ≥ 1 and dmax on Rª are all Lipschitz equivalent. 2) If X…

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Q: Let X be an infinite set equipped with the indiscrete topology T, then (X,T) is compact. * True…

A: Note: We are entitled to solve only one question at a time. As the specified one is not mentioned,…

Q: Let 7 be the Euclidean topology. Let A [0,2] and B [0,1, then Bis dopen in (AT) where 7, is the…

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Q: We define the included point topology by Tp={ UcR;U=ø or peU}. Let A = [3,5[, %3D then A is dense in…

A: See the detailed solution below.

Q: Show that close ball Y in a metfic space (X,d) is a closed set also show that if (X,d) is complete…

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

Q: Let C(X,Y) have the compact-open topology. Show that if Y is Hausdorff, then C(X, Y) is Hausdorff

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Q: Let R be equipped with the Euclidean topology T and let Y =]10,2O[. We denote by Ty the induced…

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Q: 1) In a metric space (M, d), show that both 0 and M are open sets in M. 2) In a metric space (M, d),…

A: Open sets :- Let (M, d) be a metric space then a subset A of M is open if and only if each point of…

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. biso (2) If…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Without appealing to Heine-Borel, prove that [0, 1] CR is compact and connected in the standard…

A: Please check in the next step

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: The number of elements of the coc-countable topology on the set {a, b, c, d} is O 10 8 O 2 O 4 64 O…

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Q: Let E and F be compact sets in a metric space (X, d). Show that EUF is compact using the definition…

A: 1. Let E and F be compact subsets in a metric space X,d. Let UE and UF be arbitrary open covers of E…

Q: * Let R be with the co-finite topology. If A = {1,3,5, 7,..}, then Aº R O Q O A O N O

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Q: 5-The Center of aringR dented by cent CR)is the set Cent CR) - [CERIC.X = X-C,V XER? %3D Prove that…

A: solution You just need to prove that the centre is a ring within itself. Associativity of both…

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: If U is homeomorphic to V and U is compact, then * O Vis compact V is complete V is connected O V is…

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Q: Prove by sequences that the open balls in R^n with the usual (Euclidean) metric are not compact

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Q: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

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Q: Consider IR and define Tę and E where I is the tepolagy cofinite topology and Ic is countable then…

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Q: Consider a finite set S of points in the two-dimensional plane. A point p of S is called a covidiot,…

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Q: Let X = {a, b, c, d},S = {{a, b, c}, {a}, {b, c, d}}, and T be the topology on X generated by S.…

A: We  use the definition.

Q: Let T be the Euclidean topology. Let A = [0,2] and B = [0,1[, then B is clopen in (A,T) where T is…

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

Q: Consider the following subsets of R equipped with the Euclidean topology: Зп — 1 A = {- n+1 B = Ü 1+…

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Q: (a) Prove that the metric space (Z,) is complete.

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

Q: Consider the standard topology on R, Let A=[3,7], the interior of A is Oa A O b.6 Oc. (3,7) OdR

A: Consider the standard Topology on R. Given A =[3,7] We need to find the interior of A.