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201A_F10_mt1_sol - 201A Midterm November 1 2010 Thomases...

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201A Midterm - November 1, 2010 - Thomases Name: Show all of your work, in particular note any theorems you may use and state the conditions carefully and make sure they are satisfied. Problem Possible Points Points Received 1 10 2 10 3 10 4 10 5 10 Total 50
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1. Prove that any compact metric space is separable. Proof. We know that in metric spaces the notions of compact and sequen- tially compact are equivalent. Therefore if X is a comact metric space then X is sequentially compact. By definition, sequentially compact means that there exists a finite 1 n - net for X for each n ; call it X n . Let X = S n =1 X n . Then X is countable and dense in X . Therefore X is separable.
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2. A collection F of closed subsets of a topological space ( X, T ) has the finite intersection property if ∩F 0 6 = for all finite subcollections F 0 ⊂ F . Show that ( X, T ) , a topological space, is compact if and only if every family of closed sets F ⊂ P ( X ) having the finite intersection property satisfies ∩F 6 = .
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