the collection of all close subsets of R is called the topology of R ylgn İhi

Q: For any topological space (X, T) and A C X, the set A is closed. That is, for any set A in a…

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

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

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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: * 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: Show that the dictionary order topology on the set R x R is the same as the product topology Rd × R,…

A: We have to show that the dictionary order topology on the set ℝ×ℝ is same as the product topology…

Q: Let T be the finite closed topology on a finite set X, then * O Tis the Euclidean topology O Tis the…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: (a) Consider the following topology of X = {a,b, c, d, e}: T = {X, 0, {a}, {c, d}, {a, c, d}, {b, c,…

A: In topology and related branches of mathematics, a connected space is a topological space that…

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: Which one of the following statements is true? * R with the Euclidean topology and R with the finite…

A: Answer

Q: For any collection a of subsets of X, there i ntaining a.

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Q: Let A be any set with the discrete topology. Show that A is normal.

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Q: (1) Let X be a set and tEX, let t={GCX/ t€G or G° finite }, show that whether t is a topology on X…

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Q: Consider R² with the usual topology, let A and B be two subsets of R2. We denote the closure of a…

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Q: Let X be an ordered set, and let A be a subset of X.Consider A as an ordered set, where the order on…

A: Given that X be an ordered set and let A⊂X, A is an ordered set. Y⊂X and Y is convex  in X. The…

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

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

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Q: Let X be an infinite set T be topology of the a Linfinite subsets that is on x- and let on X in…

A: Given : X be an infinite set and T be a topology on X such that all infinite subsets of X are open.…

Q: A non-empty set A in the standard topology on R so that ðA= 0.

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Q: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, 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: Q2: Let the set under consideration be N for each n EN, define u, = {n, n+1, n+2, ...} and let 1 =…

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Q: 5. If {T,} is a family of topologies on X, show that Ta is a topology on X. Is UT, a topology on X?…

A: First we will prove the result that arbitrary intersection of a topologies is a topology by showing…

Q: Which one of the following statements is true? * R with the Euclidean topology and R with the…

A: Two homeomorphic spaces share the same Topological property If one of them is compact, then the…

Q: 1. Determine if the given collection of subset of X={a, b, c} is a topology on X. a. T1 = {Ø, X,…

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

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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>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: Which one of the following statements is true? * O Rwith the Euclidean topology and R with the…

A: Which of the following statement is true?

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: a) Consider the set X = {1,2,3} with the topology T = {Ø, X,{1}, {2, 3}, {1,2, 3}}. Show that (X, t)…

A: As per our guidelines we are supposed to answer only one asked question.Kindly repost other…

Q: Which one of the following statements is true? * R with the Euclidean topology and R with the…

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Q: if

A: R is not connected if T is the finite closed topology.

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: If X = {a,b, c}, let T1 = {0, X, {a}, {a,b}} and T2 = {0, X, {a}, {b, c}} . Find the smallest…

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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 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 an infinite set and T be a topology on X. If every infinite subset of X is in T, then T is…

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Q: Which one of the following statements is true? * Rwith the Euclidean topology and R with the finite…

A: Answer

Q: Let $X$ be a set and assume that p E X. Let T, be the collection (Ø, X, and all subsets of X that…

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

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

A: As Z is with co-finite topology. Every open neighborhood of an infinite set must contain infinitely…

Q: Let (X. T) be a T, topological space and A a nonempty subset of X, if x is a cluster point of A…

A: Given: Let (X,T) be a T1 topological space and A a non-empty subset of X, if x is a cluster point of…

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: Show that the dictionary order topology on the set R × R is the same as the product topology ℝ_d × ℝ…

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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 X = {a, b, c}. Is the collection {X, Ø, {a}, {a, b}, {c}} a topology on X?

A: Here, the given set is X=a,b,c and collection is X,∅,a,a,b,c.

Q: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

A: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

Q: Consider X = {a1, a2, ..., an} Can a topology of X with 3 open and one with 4 open be equivalent?

A: We have to verify the given statement.

Q: Show that the subset (a, b) of R is homeomorphic with R.

A: Recall that every nonempty open interval (a,b) is homeomorphic to the nonempty interval (c,d).

Q: Let N be with the discrete topology If A=(1,3,5..), then Bd(A)=

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Q: Which one of the following statements is true?* R with the Euclidean topology and R with the…

A: Which one of the following statement is true