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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A: Explanation of the answer is as follows. As per the rules I am doing first three sub parts in…

Q: (1) Ø° = Ø and M° = M. (2) If SST, then S° ST°. (4) S is open if, and only if, S = S

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A: Given a question

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A: First and last one has counter examples see the step 2 and second one is true

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Q: A ° = A – JA

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A: See the attachment.

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Q: Ans 7 only

A: To establish the required topological properties related to interior and exterior points.

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A: Given below the detailed solution

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A: This is a problem of Topology.

Q: PLS HELP ASAP ON ALL ASKED QUESTIONS PLS

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A: As per rule, allowed to answer first question and post the remaining in the next submisison.

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Q: Theorem (Axioms of an interior): Let (M, d) be a metric space and S, T M. Then (5) S°° = S°.

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Q: pls explain to me step by step and pls write clearly. thanks

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