Theorem 2.15. Let (X,T) be a topological space, and let U be an open set and A a closed subset of X. Then the set U – A is open and the set A – U is closed.

Q: 3.2 Prove that in any metric space (S, d) every closed ball S,xo is a closed set.

A: Let (S, d) be a metric space. Consider a closed ball Srx0 with both center x0∈S and radius r>0…

Q: Theorem 4.1. A space (X,T) is T¡ if and only if every point in X is a closed set.

A: Given a topological space X,T and X is a T1-space if and only if for every pair x, y of distinct…

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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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: 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: - Let (X, t) be a topological space and A, BCX, then 1- A° 2 A- b(A) 2- A° # A- b(A) 3- A° = A –…

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Q: Theorem 6.2. Let C be a compact subset of Rstd· Then C has a maximum point, that is, there is a…

A: We start the proof by assuming there are no such maximum element in C.

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

A: To check: Whether the statement is true or false: Given: The statement is "The collection of all…

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: Theorem 4.4.8. Let (S, d) be a metric space. A set A is open if and only if its complement R\A is…

A: Given that a<b in ℝ The objective is to determine whether the given sets are open or not.

Q: Theorem 2.26. Let A be a subset of a topological space X. Then p is an interior point of A if and…

A: First well show that, if there existsan open set U with P∈UCA, thenP will be an interior point of…

Q: Theorem 7.3. Let X c Y be topological spaces. The inclusion map i : X → Y defined by i(x) 3D х is…

A: Suppose U is an open set in X. Its inverse image is then i−1(U) = {x ∈ A : i(x) ∈ U}          = {x…

Q: Theorem 4.8. A topological space X is regular if and only if for each point p in X and ppen set U…

A: given p ∈ X, and a neighbourhood U of p, there is an open set V ⊂ X suchthat p ∈  V ⊂ U. Consider…

Q: Let (X,T) be a topological space Property C=P.C.  A subset A of x has P.C If it's subset of the…

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Q: 1. (a) Define what a linear subset of a projective space is.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: 1. Prove that for all sets X, Y , if XU Y = Xn Y, then X = Y. %3D

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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: A and B are open subsets of X with Cl(A) = X = Cl(B).

A:

Q: Question 10. Prove or disprove. Any infinite subset A of a discrete topological space (X,r) is…

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

A: the collection of all close subsets of R is called the topology of R

Q: Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only…

A: Given: X,d is a metric space, x∈X and A⊂ X is a non-empty set. To Prove: dx,A=0 if and only if every…

Q: 2. Find the complement CC of the set C with respect to the space C if (a) C = {x :0 < x < 1}, C = {x…

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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: 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 X and Y be topological spaces and fix some yo E Y. Show that the map f: X → X x Y defined by f…

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Q: Exercise 2.27. Show that a set U is open in a topological space X if and only if every point of U is…

A: Let, U is an open set in a topological space X. Then, U⊂U  So, U⊂int(U) and int(U)⊂U Hence, U=int(U)

Q: 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.…

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Q: Theorem 4.9. A topological space X is normal if and only if for each closed set A in X and open set…

A: Consider X,I be a normal space and U⊂X be an open neighbourhood of a closed set A⊂X, then A⊂U To…

Q: 2.5 For each of the following properties, give an example consisting of two subsets X, Y CR', both…

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Q: Theorem 2.3. A set U is open in a topological space (X,J) if and only if for every point x€ U, there…

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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: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

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

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

Q: Let (X,T) be a topological space such that every subset of X is closed, then a. (X,T) is an…

A: (X,T) is a topological space such that every subset of X is closed

Q: a) Consider the set X = {1,2,3} with the topology t = Show that (X, t) is a topological space. (0,X,…

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

Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

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Q: Theorem 6.17. Every compact subset C of R contains a maximum in the set C, i.e., there is an m E C…

A: To prove that, every compact subset C of ℝ contains a maximum in the set C, that is, there is an m∈C…

Q: consider R with the usual topology then a subset A={1,0.5,..} is: empty or closed or open or none…

A: Given R with the usual topology and given a subset  A=1, 12, 13,.....

Q: Prove the following: Let X denote the set {0,1} with the discrete topology. Let Y = X" = || X; ieZ+…

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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 and Y be two discrete spaces, then * X is homeomorphic to Y if and only if X and Y have the…

A: X and Y are two discrete spaces then X and Y   The topology is  discrete then for x∈X  is an open…

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

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Q: Theorem 8.11. For topological spaces X and Y, X ×Y is connected if and only if each of X and Y is…

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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: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

A: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

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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