# question related to outer measure and pseudometric.

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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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 $m^\ast(O \setminus E) < \varepsilon$.