Give an example of an open cover of the segment (0, 1) which has no finite...

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Give an example of an open cover of the segment (0, 1) which has no finite cover. 1) If A and B are disjoint closed sets in a metric space X, prove that they are separated 2) Prove the same for disjoint open sets. Let X be a nonempty infinite set and d a metric on X. Prove that the following functions on X are also metrics : a) f(a,b) = min (1,d(a,b)) b) g(a, b) = d(a.b) 1+d(a,b) where a, b X are also metrics on X. If A and B are compact subsets of a metric space X, show that AUB is compact. Give an example of an open cover of the segment (0, 1) which has no finite cover. 1) If A and B are disjoint closed sets in a metric space X, prove that they are separated 2) Prove the same for disjoint open sets. Let X be a nonempty infinite set and d a metric on X. Prove that the following functions on X are also metrics : a) f(a,b) = min (1,d(a,b)) b) g(a, b) = d(a.b) 1+d(a,b) where a, b X are also metrics on X. If A and B are compact subsets of a metric space X, show that AUB is compact.

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An example of an open cover of the segment 01 that has no finite subcover is the collection of open ... View the full answer