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Srinath Best Answer
Consider the element 5 € N, the set of all natural numbers. Choose € = 1/2. Then (5 - 0.5, 5 + 0.5) which is equal to (4.5, 5.5), is NOT a subset of natural numbers, even though 5 belongs to that open interval. In order to be a neighborhood of N, it has to be a subset of N as well.