Theorem (Axioms of an interior): Let (M, d) be a metric space and S,T CM. Then: (1) 0 =0 and M = M. (2) If SST, then S' ST. (3) S is the largest open set in M that contained in S. (4) S is open if, and only if, S = S. (5) S" = S°. (6) (Sn T)'= S' n T". (7) In general, S UT S (SUT), but (S U T) S UT.

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