8. Give an example to show that the arbitrary intersection of open sets is not an open set. 9. Give an example of a set which neither closed nor open. 10. Show by an example that a nonempty proper subset of 5 in the metric space (S, d) can be both open and closed.

Q: i. PROVE THAT THE EXCLUDED POINT TOPOLOGY ON ANY SET WITH AT LEAST 2 POINTS IS NOT HAUSDORFF. ii.…

A: Definition: A topological space X, τ is said to be Hausdorff if for every x, y∈X there exits U, V∈τ…

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

A: We need to show whether the given statement "If a set has empty interior, then so does it's closure"…

Q: 13. The interior of a set coincides with the interior of its closure..

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let A and B be subsets of R which are bounded above. Show that sup (AnB) ≤ min {sup (A), sup (B)}…

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Q: 26

A: The objective is to find an infinite collection of closed sets whose union is not closed.

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Q: Statement 6.5. Let A be a finite set. If BC A, then B is finite.

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Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: Explain with an example of a topological space which is singleton that is open but not close and…

A: Example in refer of not  open but closed .Take , Here A is singletone set which is closed but not…

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Q: Exercise 2.7. Give an example of a topological space and a collection of open sets in that…

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Q: Determine whether the set spans all of M2. Justify your answer with your work and/on written…

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Q: dy + y cos(x) = 6 cos(x) dx

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Q: Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact

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Q: Find all the values of x such that the given series would converge. − 1) n7n xn (√n+11) ∞ n=1 The…

A: Given series is                                                                  We have to find the…

Q: A, B be subsets of a metric space. Show that A U B = B and that A B. Give an example to show that An…

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Q: True or false 1.The intersection of two sets is the set of all the equal elements that belong to…

A: Since you have not mentioned the questions to be solved, so we are providing you solutions for first…

Q: Prove that every subset E of R is the union of some family of closed sets. Can every subset E of R…

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