the collection of all close subsets of R is called the topology ofR 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: Let X be an infinite set equipped with the finite closed topology. A finite subset of X is a. closed…

A: The "finite closed topology" describes the closed sets. A subset U is open if and only if U=∅ or X∖U…

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: Let X = {a, b, c, d, e, f, g} and B = {{a,c, e, f}, {b, c, d, f,g}, {c}, {f}}. Show that B is a base…

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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 (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 A be any set with the discrete topology. Show that A is normal.

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

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Q: I? B is a family of subsets of X Such that: (2) If B, ,BzE/ß them B,nB, EB Then B is a base for a…

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Q: Prove that the complement of a dense open subset of R is nowhere dense in R

A: Prove that the complement of a dense open subset of R is nowhere dense in R

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 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 A1,·.., Am be compact subsets of R. Prove that Aı × · … × Am is a compact subset of Rm.

A: Given, A1,A2, ... , Am are compact subsets of R. We prove A1x A2 x ... x Am is a compact subsets of…

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: Prove that T is the discrete topology for X iff every point in X is an open set

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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 > a, a∈R ∪ x : x ≤ b, b∈R     To…

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: 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: If A and B are nonempty bounded subsets of R with A C B, prove that sup A < sup B.

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Q: In R with the cofinite topology, every preopen set is open

A: As we know that , In ℝ with discrete topology of every semi open set is open.

Q: Let (X.J) be a topolgical space and {C1, C2 Cn} be a finite collection of compact subsets of X.…

A: Given Let (X,I) be a topological space and C1,C2,......Cn be a finite collection of compact subset…

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: (c) Prove that only Ø and R are open subsets of R

A: A subset Uof a metric space (M, d) is called open if, for any point x in U, there exists a real…

Q: df. Every nonempty proper subset of R is either open or closed.

A: False

Q: (a) If {Ta} is a family of topologies on X, show that N Ta is a topology on X. Is UTa a topology on…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: 1. Let T and S be topologies in X. Show that SnT is also a topology in X.

A: 1.

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

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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: Every proper simply connected oper equivalent to an open subset of the

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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: (b) Prove that the complement of a dense open subset of R is nowhere dense in R..

A: Prove that the complement of a dense open subset of R is nowhere dense in R.

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

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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: Suppose A is a nonempty subset of R. (a) Is it true that Là=L? Prove or disprove. (b) Let iso(A)…

A: Given that A is a non-empty subset of ℝ. (a) To prove that                      LA¯ = LA

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

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Q: Let {Fa}aeA be a collection of closed subsets of R. (a) Prove, from the definition of closed, that…

A: Given that Fαα∈A is the collection of closed subsets of ℝ. The objective is: To prove that ∩α∈AFαis…

Q: 5. Find the boundary of the following subsets of R in the standard topology. (1) B = {x € R | x² >…

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

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Q: If K is a compact subset of RP and z is a point of RP, then the set K, = {r+y :y € K} is also…

A: Given that K is a compact subset of ℝp and x is a point of ℝp. The objective is to show that…