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Im currently reading the book of Steven R. Lay's "Analysis with an Introduction to Proof, 3rd ed.". According to his book, if a subset S of ℝ contains all of its boundary then it is closed. But i find this wrong since if we consider S={xεQ;0≤x≤2}, then it can be shown that S contains all of its boundary points (using the fact the Q is dense in ℝ), but it is not closed since the closure of S is the interval [0,2] which is not equal to the set itself. am i correct?

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# Definition of closed sets

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