1) Let X be any non-empty set and xo E X. a) Prove that T = {X}U{AC X : xo ¢ A} is a topology on X. b) What is the closure in (X, T) of any subset A of X.

Q: Let O be the collection of intervals Ia = (a, ∞) where a R along with I = 0 and I-∞ = R. Does this…

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Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

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

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Q: 1. Let X = {a,b,c} and B={{a,c}.{b,c}}cP(X). Show that B cannot be a base for any topology r on X.

A: According to the answering guidelines, we shall solve first question only. If you want others to be…

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: Let R be equipped with the Euclidean topology T and let Y =]10,20[. We denote by Ty the induced…

A: According to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Let (X, I) be a topological space and suppose that A and B are subsets of X such that ACB. Prove…

A: Let x ∈A°. Then by definition, there exists an open set U  such that  x∈U⊆A.Since A⊆B hence x∈U⊆B ,…

Q: Let X =[0,2).Define r= {[0, a):0sa s2). 1. Show that r is a topology on X. 2. Give an example which…

A: Let X be any set and ζ be the collection of subsets of x then ζ is called as topology if it contains…

Q: 4) Consider the collection of intervals on the real line: B = { (a, b) | a 1}. Is this a basis for…

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Q: Which of the following is a topology on R? {UCR: U is infinite}U{U<R: U is countable} O {[a, b): a,…

A: We will use the definition of Topology.

Q: In a Cofinite topology of X the collection of subsets a) r = {AcX|A ±Ø or A is finite} b) r = {Ac X|…

A: Let X be a non-empty set, then the collection of subsets of X whose complements are finite along…

Q: Let d and e be metrics on a set X such that for each ball Ba center at pE X there exists a ball Be…

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Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: Open in subspace Y means it can be written as open set in X intersection Y.

Q: Let X be an infinite set and T a topology on X. If every infinite subset of X is in T, prove that T…

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Q: Let A be a subset of a topological space X. Show that bd(A) = Ø iff A is open and closed in X.

A: We need to prove the statement.

Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: See the detailed solution below.

Q: Let X = R and s be a collection given by ( = {0}U{0cX:0° <x}. (a) Show that S forms a topological…

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

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: Theorem 2.28. Let A be a subset of a topological space X. Then Int(A), Bd(A), and Int(X – A) are…

A: Given that A is a subset of the topological space X. We have to prove that IntA, BdA and IntX-A are…

Q: Consider the set A=(x:x>a, a E R}U(x:xsb, bE R). Find the topology on R which has A as subbases.

A: Given :               Given a set                   A = x : x &gt; a, a∈R ∪ x : x ≤ b, b∈R     To…

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

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Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: See the detailed solution below.

Q: Consider the set A=(x:x>a, a ER)U(x:xsb, bER). Find the topology on R which has A as subbases.

A: given a set  A=x:x&gt;a,a∈R∪x:x≤b,b∈R to find the topology on R which has A  as subbases

Q: Let τs and τ be the standard topology and the countable complement topology on R, respectively.…

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Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

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Q: Prove the following sets are not compact by finding an open cover that does not have a finite…

A: a) Given-  A set A=−1,1. Explanation- Consider a set Xn=-2,1-1n for n=1,2.... Then clearly,…

Q: Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set…

A: Given that A is a subset of the topological space X and p is a point in X. Consider a sequence…

Q: Let R be equipped with the standard topology. Define q : R? → R, q((x1, x2) : Jx1, if x1 2 x2 |X2,…

A: Question: The function q:ℝ2→ℝ defined by qx1,x2=x1, if x1≥x2x2, if x1&lt;x2 where ℝ2 and ℝ are…

Q: Let T be a topology on X. Assume that T is Hausdorff and let x € X. (1) Show that {x} (and hence…

A: let T be a topology on x. T is a Hausdorff space

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

A:

Q: In a Cofinite topology of X the collection of subsets is: a) r = {AcX|A±Ø or A is finite} b) r = {Ac…

A: Let X be a non empty set , then the collection of subsets of X whose compliments are finite along…

Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A:

Q: Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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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: Let X be a topological space. Construct a topological space Px and a continuous surjection 0: X→ Px…

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Q: Let X be a set with more than one element. (X,t) is a topological space such tha Vxe X,{x} €T then…

A: Given a set X with more than one element.

Q: Let A CX with the discrete topology. Prove that ô(A) = Ø.

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Q: Let X be an infinite set with the countable closed topology T={S subset of X; X-S is countable}.…

A: Connected means it cannot be written as union of disjoint open sets

Q: Let X be a topological space and A be a subset of X. Prove that (A°)° = Ac and (A) = (A°)°.

A: Let A be a subset of a topological space X. Let the interior of A be denoted by A∘. Let x∈A∘c. Then…

Q: Let X be an infinite set and T be a topology on X. If every infinite subset of X is closed, then T…

A: Given that Let X be an infinite set and T be a topology on X.If every infinite subset of X is…

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: 1) Let X be any non-empty set and xo E X. a) Prove that T = {X}U{ACX: 2o ¢ A} is a topology on X. b)…

A: Topology means we have to prove 3 conditions then we can tell that space is Topology.

Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: Let R be a  equipped with the Euclidean topology T

Q: f A∩B≠∅ and A∩Bc≠∅ then show that A∩X\B̅≠∅ and A∩B̊≠∅

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Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: Let R be with Euclidean topology

Q: Let X = R, and let T consist of 0, R, and all intervals of the form (a, b) where a, b e R and a < b.…

A: A topological space is an ordered pair {\displaystyle (X,\tau ),} where {\displaystyle X} is a set…

Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: here option (c) is true because

Q: 1. Let X be any nonempty set and let p E X, Prove that : E, {G X:p¢ G}U{X} is a topology on X.

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

Q: Let R be equipped with the standard topology. Define q : R? → R, x1, if x1 2 x2 X2, if x1 < x2…

A: See the attachment

Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: Induced topology