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# hw2 - 7 A collection V α of open sets of X is said to be a...

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Homework 2 – Math 118A, Fall 2009 Due on Thursday, October 15th, 2009 1. Give an example of an open cover of (0 , 1) which has no finite subcov- ering. 2. Regard Q , the set of all rational numbers, as a metric space, with d ( p, q ) = | p - q | . Let E be the set of all p Q such that 2 < p 2 < 3. Show that E is closed and bounded in Q , but E is not compact. Is E open in Q ? 3. Construct a compact set of real numbers whose limit points form a countable set. 4. (a) If A and B are disjoint closed sets in some metric space X , prove that they are separated. (b) Prove the same for open sets. (c) Fix p X , δ > 0, define A to be the set of all q X for which d ( p, q ) < δ , define B similarly, with > instead of < . Prove that A and B are separated. (d) Prove that every connected metric space with at least two points is uncountable. 5. Let ( X, d ) be a metric space. Is it true that for any r > 0, and for any p X , B r ( p ) = { q X | d ( p, q ) r } ? (1) 6. A metric space is called separable if it contains a countable dense set. Show that

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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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