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

Q: 11. If a set has empty interior, then so does its closure. 12. The closure of a set coincides with…

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Q: 5 = { [²] [3²] [37] 3 a) Determine if S = is an orthogonal set. b) True or False. If A = ²24 then A…

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Q: 2. Construct two metric spaces X and Y which are homeomorphic, but X is complete and Y is not…

A: 2. Construct two metric spaces  and  which are homeomorphic, but  is complete and  is not complete.

Q: Let X = {1, (b) xm} and Y = {y₁, Yn} be finite sets. Determine the number of one to one functions…

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Q: Suppose (X, d) is a metric space and E a non-empty subset of X. Prove that E is disconnected if and…

A: Use definition of disconnectedness mentioned below.

Q: Solve number 1

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Q: 1- In a metric space (X, d), if A and A2 are open subsets of X then A1 U A2 is open in X.

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Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. biso (2) If…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

Q: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: 5. In any metric space (M, D), prove that D(a1, an) ≤ D(a₁, a2) + D(a2, a3) + + D(an-1, an), for a1,…

A: Given M,D is a metric space.   The triangle inequality holds in a metric space, which means for…