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

Q: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

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Q: Let A and B be subsets of topological spaces X and Y, respectively. Prove that CI(A × B) Cl(A) ×…

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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: A metric space (X , d) and a collection of open sets U such that UEU is not open.

A: Let X, d be a metric space. Let U be the collection of open sets in the metric space X, d. That is,…

Q: If S is compact and D is in S is a closed set, then D is compact. Use the definition of…

A: Let S is compact and D be a closed subset of S. We are to prove that the set D is compact in S. Now…

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

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Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

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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: * 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: Let X be an infinite set. (a) Show that is a topology on X, called the finite co со

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Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

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Q: Let (X,t) be a topological space, A, B C X. Then A' UB' = (A U B)', A'n B' c (A N B)' Select one: O…

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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.…

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Q: Let

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Q: If X is a normed space then the open ball and closed ball is a Example: convex set

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Q: Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset of X such that…

A: Let C be a connected subset of the topological space X.Also given that D is a subset of X such that…

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: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify tha the quotient…

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Q: Let (X, T) be a topological space and A, B be a subsets of X then: اختر احدى الدجابات int(A - B)=…

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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: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify that the quotient…

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Q: Let X = {a,b,c} prove that topological space on the set X T = {X. Ø, {b}, {a,b}} be a

A: Let X={a,b,c} be a set. To show that T={X,ϕ,{b},{a,b}} be a topological space on the set X.

Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

Q: 5. Let X be a set and p E X. b) If 72 = {A C X : p ¢ A}U{X}is a topology on X, then determine…

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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: Prove the following. Let J be any set, and let {Xa : a E J} be an indexed family of connected…

A: It is given that Xα:α∈J is an indexed family of connected topological space that is each component…

Q: Let lx, T) be a topological Space and A, B S X- if A is open, then show that An CIIB) s CLLANB). if…

A: Given X, τ be a topological space and A, B⊆X. Suppose, A is open then An arbitrary x∈A∩clB implies…

Q: If x is a limit point of a subset A of a metric space X, then sh

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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: If A is a subset of a metric space X, then show that A = (x: each neighbourhood of x intersects A}

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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: Exercise 8. Prove the following properties of neighborhoods: Proposition. If x is a point of the…

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Q: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify that the quotient…

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Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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Q: the following subsets in an indicated metric space determine the interior, boundary points and…

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Q: Amongst the various possible products of four topological spaces A, B, C and D, show that (A × (B ×…

A: We shall solve this in next step

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: For any metric space (X, d) and x EX, the singleton set (x) is closed. Select one: OTrue O False

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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 (X7) be a topological Space and Show that A is open and closed m (xx) iff %3D

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

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

A: False

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

Q: If (X,, T,), (X, T2) are two topological spaces, and A,, A, are open sets in X,and X, respectively,…

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