OShow that the set (1,2] is not compact 2Show that the set [1,2) is not compact 3Show that the set (1,2) is not compact

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A: to prove 3.1 first we will prove that infinite union of open sets in a metric space ( S, d ) is open…

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Q: 4. Let E1, E2 be two compact subsets of R'. Use the Heine-Borel Theorem to prove that EUE2 is also a…

A: As you have asked more than one question, we Shall answer first one only. For others please post…

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A: Thanks for the question :)And your upvote will be really appreciable ;)

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Q: Which one of the following statements is true? * R with the Euclidean topology and R with the finite…

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Q: 100. In a topological space, which of the following statements is correct? Sisianoge (A) a compact…

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Q: 3.1 Show that an infinite intersection of closed sets F, k = 1,2,3, . (S, d) is a closed set. in a…

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A: Consider the following statements: (i) If A is not compact then there are infinitely many open…

Q: 10. Two sets are called disjoint if they have no elements in common, i.e., if their intersection is…

A: Note that two sets are called disjoint if they have no elements in common, that is, if their…

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Q: al let (xt) betopology space and BCY then B is closed in y iff there is closed set CinX st B-COY

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A: For the solution follow the next steps.

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Q: True or False: Consider the subsets (0, 1) and (2, 3) of R, the set of all real numbers with the…

A: The given statement is  Consider the subsets (0, 1) and (2, 3) of R, the set of all realnumbers with…

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Q: 6. For A, B, subscts of R, define A+B= {a+b:a€ A, bE B} and A B= (ab:a € A,bE B} (a) For A=…

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: First option is correct

Q: 3.1 Show that an infinite intersection of closed sets Fa, k = 1, 2,3,.., in a metric space (S, d) is…

A: The given is related to Metric spaces. The solution is explained below.

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Q: a. Find a bijection between Zt and the set {1,1/2,1/4, 1/8, 1/16, 1/32, ...}. b. Show that if A is…

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Q: If A is finite, then a) there is an injection of A with a proper subset of itself. b) there is a…

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Q: 1. Prove that (1, 2] is not compact using the definition.

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: Given: A property is said to be a topological property if it is preserved by homeomorphism. Suppose…

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Q: space. connected subsets of X such that C; is non-empty, then C = C; is also

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Q: 2. Prove Lemma: For all sets A, B, and C, A x (BnC) = (A × B)n (A × C)

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Q: (a). Find a set S and a bijection between S and a proper subset of S. (b). What condition on S…

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Q: 3. Prove that if K1 and K2 are compact sets in R, then K1UK2 is compact in R using (a) the…

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Q: 3.1 Show that an infinite intersection of closed sets F, k 1,2, 3,..., in a metric snace (S, d) is a…

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Q: Please solve part (a).

A: Hi! We are answering only the first two problems. If you need the solution of rest questions, then…

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A: Given that A is non empty set which is bounded below by 3. Given that B=x∈R|∃a∈A,b=1a The objective…

Q: 8. Let X be a set and ß be a basis for a topology T on X. Then T equals the collection of all unions…

A: Given: X be a set and β be a basis for topology τ on X. To prove: τ equals the collection of all…