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    Sets & Proofs

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    Let S be a subset of the metric space E. A point p (element of ) S is called an interior point of S if there is an open ball in E of center p which is contained in S. Prove that the set of interior points of S is an open subset of E (called the interior of S) that contains all other open subsets of E that are contained in S.

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    Solution Preview

    Let K is the set of all interior points of S.
    First, I show that K is an open subset of E.
    For any x in K, since x is an interior point of S, by definition, we can find an open ball B(x;r) that is contained in S, where
    B(x;r) = {y in S: |y ...

    Solution Summary

    Sets and proofs are examined. The expert proves that the set of the interior points of S is an open subset of E.