Prove the following: Suppose that {X; : i e I} and {Y; : i e I} are indexed families of topological spaces. Suppose that for each i E I, the spaces X; and Y; are homeomorphic. Then [I;e1 X; and [I;e, Yi are homeomorphic. lieI LieI

Q: 5. If G1,G2 be two given closed sets in a metric space (X, d) prove that: G1 U G2 is closed.

A: We have to prove given result:

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

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Q: (1) Determine if each of the following statements is true(T) or false(F). 1- In a metric space,…

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

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

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Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

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Q: Prove the following: Suppose that {X; : i e I} is an indexed family of topological spaces. Suppose…

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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: Prove the following: Suppose that {X; : i € I} and {Y; : i E I} are indexed families of topological…

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Q: 6. Let (X,T) be a topological space such that every subset of X is closed, then a. (X,T) is an…

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

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Q: Hausdorff space.

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Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

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Q: in the discrete metric space: 1. All collections are clo

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

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

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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: A discrete metric space X is separable if and only if X is countable

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

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A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

Q: 5. State whether the following is true or false m) A subset I of a metric space R with the usual…

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