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Unformatted text preview: 7. A collection { V α } of open sets of X is said to be a base for X if the following is true: ±or every x ∈ X and every open set G ⊂ X such that x ∈ G , we have x ∈ V α ⊂ G for some α . In other words, every open set in X is the union of a subcollection of { V α } . Prove that every separable metric space has a countable basis. 8. Let X be a metric space in which every inFnite subset has a limit point. Prove that X is separable. (See problem 24 on page 45 in Rudin’s book for a hint). 1 2 9. Prove that every compact metric space K has a countable base, and that K is therefore separable. (See problem 25 on page 45 in Rudin’s book for a hint). 10. Let X be a metric space in which every inFnite subset has a limit point. Prove that X is compact. (See problem 26 on page 45 in Rudin’s book for a hint)....
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 Fall '08
 Stopple,J
 Math, Topology, Metric space, RK

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