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Unformatted text preview: 7. A collection { V } 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 G for some . In other words, every open set in X is the union of a subcollection of { V } . 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 Rudins 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 Rudins 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 Rudins book for a hint)....
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This note was uploaded on 12/27/2011 for the course MATH 118a taught by Professor Stopple,j during the Fall '08 term at UCSB.
 Fall '08
 Stopple,J
 Math

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