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I want to show that if O is collection of open subsets of (0,1) what is the closure of O in the associated metric space of equivalence classes? The metric associated with this collection is pseudometric which is equal to outer measure of symmetric difference of two subsets of (0,1).

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1 Answer 1

Hint: For any $E \subset (0,1)$ and $\varepsilon>0$ by definition of outer measure we can find an $O$ such that $E \subset O$ and $m^\ast(O \setminus E) < \varepsilon$.

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I thought on these lines but infact I want to make a sequence of open sets that converges to limit in (0,1), as I want to show that this collection is complete or not? –  Athar Raheel Ahmad Nov 13 '12 at 20:13
    
Well, if you prove my hint then you've shown that the open sets are dense. If they're dense, then what does their closure have to be? –  JSchlather Nov 13 '12 at 20:16
    
well Jacob, I started with the sequence of open sets contained in (0,1) and constructed the sequence in such a way that it goes with the conditions of nested theorm, now my only concern is that can I show this sequence to be a convergent sequence? If so then I can go to the fact of completeness through density and closure definitely. What's your opinion ? –  Athar Raheel Ahmad Nov 14 '12 at 15:40

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