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

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Q: Show that R is a Banach space.

A: Banach space: A complete normed linear space is called Banach space. Given: ℝ3=x(1, x(2), x(3)) :…

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A: We will prove the given statement.

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A: For a metric space we know that the for a compact set every open cover has a finite subcover. So…

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Q: For a normed space X and a closed convex subset A of X. Prove that (º )nÎ= Â. A

A: Let, be a normed space and be the closed convex subset of As, we know that norm on is a function…

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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…

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A: As per bartleby guidelines for more than three subparts in a question only first three should be…

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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: 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: For any subset A of a topological space X, we define the boundary of A to be aA = AnX\A. a) Show…

A: (a)

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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: Suppose X and Y are normed spaces and X + 0. Prove that BL(X, Y) is a Banach space with the operator…

A: Given, X and Y are normed spaces and X ≠ 0. We have to prove that BLX , Y is a Banach space with the…

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

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Q: Let ( Xr d be ame tric space,show that exist atopology on X induce b

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Q: If M and N are metric spaces, the Cartesian product M x N is again a netric space with the distance…

A: We have given that , M and N are metric spaces. We have to show that , M × N is again a metric space…

Q: Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

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Q: Let X be a finite dimensional norm space. Then prove that M = {x € X |||x|| <1} is compact.

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Q: let C be a compact subset in a Hausdorff topological space (X,T) and b∈X a point. If b∉C then prove…

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Q: ) Let X be a finite dimensional norm space. Then prove that M = {x EX |||x || <1} is %3D compact.

A: Sufficient to prove that M is closed and bounded

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A: Question: The function q:ℝ2→ℝ defined by qx1,x2=x1, if x1≥x2x2, if x1&lt;x2 where ℝ2 and ℝ are…

Q: Prove: If F is a closed subset of a compact space X, then F is also compact.

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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 (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

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Q: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

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

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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: Let {Xa | a E A} be a collection of Hausdorff topological spaces. Prove that IαEA Xa is Hausdorff…

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Q: Prove that Let X be a metric space such that X E1 U E2. If E1, E2 are complete, then X is complete.

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Q: If X is an inner product space and A = {0}, then A = %3D %3D O X O A

A: A⊥ = X

Q: Let X be a topological space and A be a subset of X. Prove that (A°)° = Ac and (A) = (A°)°.

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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: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

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Q: Show the following result: Theorem. Let (X, d) be a metric space, A C X, and let u: A → R be an…

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Q: Let R be equipped with the standard topology. Define q : R? → R, x1, if x1 2 x2 X2, if x1 < x2…

A: See the attachment

Q: Let (X, Tx), (Y, Ty) represents topological spaces such that X = {a,b,c,d}, Tx = {X. Ø, {a}, {a, b},…

A: CONTINUOUS FUNCTION: If the inverse image of every open subset of Y is open in X, then the function…

Q: Let X be a discrete spaces then * X is never homeomorphic to R O X is homeomorphic to R if and only…

A: Fourth option is correct.