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

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: Theorem 3.7: A subset A ofa metric space (X, d) is closed if and only if /.contains all its limit…

A: We have to prove that A is Closed if and only if A contains all its limit points. Note : In proof i…

Q: 2.4.5. Let X be a metric space. Given x E X, prove that {x} is a closed subset of X.

A: Let (X,d) be a metric space to show that {x} is closed sunset of X.

Q: Question 7. Let (X,r) be a topological space and A, BCX, Determine whether A-B2A-B or not.

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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

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

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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 2.9. Suppose p ¢ A in a topological space (X,T). Then p is not a limit point of A if and…

A: Given that, Let (X, T) be the topological space. p is the not element of set A. We have to show…

Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: Theorem 3.7: A subset A of a metric space (X, d) is closed if and only if/.contains all its limit…

A: A subset A of a metric space (X, d) is closed iff A contains all its limit points.

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: 3. Let (X, T) be the topological space defined in question 2., then the closure of {2} is a. X b.…

A: It depends on the given topological space. Here topological space is not specified. If it's usual…

Q: Let X be a discrete spaces then O X is homeomorphic to R if and. only if X is countable O X is…

A: I have provided a solution in step2

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

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Q: Theorem 7.36. Let X, Y, and Z be topological spaces. A function g : Z → X × Y is continuous if and…

A: To prove: Let X, Y, and Z be topological spaces. A function  g : Z → X×Y is continuous if only if…

Q: Let X and Y be two topological spaces. X is connected and Y is disconnected are they homotopy…

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Q: Question 4. Let A be a subset of a topological space (X,r). Prove that the following statements are…

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Q: Theorem 6.3. If X is a compact space, then every infinite subset of X has a limit point.

A: We have given that, X is Compact space.

Q: The set [0, 1] with the discrete metric is compact. O True O False

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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: Question 1. Let (X,7) be a topological space and let F be the collection of all closed sets, Show…

A: Closed Sets: A subset of a topological space is said to be closed if the set is open De Morgan's…

Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: The objective is to choose the correct option: Let X be a discrete space then  a) X is homeomorphic…

Q: QUESTION 9 Suppose metric space X is not path-wise connected then X is not connected. True False

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Q: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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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: 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: Show that any infinite subset A of a discrete topological space X is not compact.

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Q: QUESTION 13 A subset I of a metric space R with the usual metric is compact if and if only it is an…

A: In usual metric space a set is compact iff it is closed and bounded.

Q: Question No 1 (8). Prove that the discrete topology on an uncountable set does not Satisty the…

A: Note- Let X be a topological space, then X satisfies the second axiom of countability or is second…

Q: 1. Let M, d be a metric space with A C B C M. Suppose that B is totally bounded. Prove that A is…

A: Let (M, d) be a metric space and let A be a nonempty subset of M. The diameter of A is defined as…

Q: Theorem 7.36. Let X, Y, and Z be topological spaces. A function g : Z → X × Y is continuous if and…

A: The given theorem is Let X,Y and Z be topological spaces. A function g : Z→X×Y is continuous if and…

Q: Question 2 Prove directly (i.e. from the definition of compactness) that if K is a com- pact subset…

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Q: (d) Every continuous map on any metric space is an open map.

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Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: Given that, X is a discrete space. To be homeomorphic X has to be of same cardinality as ℝ and so X…

Q: Every closed subset of a connected metric space is connected. O True O False

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Q: Question 3. all its cluster points. Prove that a subset of a metric space is closed if and only if…

A: We have to prove that a subset of a metric space is closed if and only if it contains its cluster…

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: (2) Let (X,r)be a topological space, If for every A open set, and x E A", 3 Ux and Vy open sets,…

A: A topological space X is Hausdorff if for each pair of distinct points p and q in X , there are…

Q: Question 2* Which one of the followings is compact a. A discrete space with at least to points. b.…

A: Given: Statements: (a) A discrete space with at least two points is compact. (b) Any set with the…

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: Question 7. Let (x,r) topological space and A, BcX. Determine be a whether A-B2A-B or not.

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Q: Theorem 2.20. For any set A in a topological space X, the closure of A equals the inter- section of…

A: Let A  be a set in the topological space X. To prove that the closure of A  is the intersection of…

Q: Every closed subset of a connected metric space is connected. True False

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Q: Question 8. Let A = {10,11,12} be a subset in the usual topological space. Find A'. Question 9.…

A: Since multiple questions are given in the question, the solution to the first question is given…

Q: Let E,F ⊆ X,d be compact subsets of a metric space X. Just using the definition of compactness…

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Q: Question 9. Prove or disprove. If (X,r,) and (X,r,) are T,-spaces, then (X,7,nt,) is also a T,-…

A: -space: A topological space is -space if it satisfies the following axiom of Frechet : "If x and y…