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In a topological space, if all elements of a subset are isolated, will the subset have any limit point? If it does have a limit point, then the limit point must of course not belong to the subset. Thanks and regards!

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up vote 5 down vote accepted

Sure, it can. Look at the set $\{\frac{1}{n}\}$, where $n$ ranges over the positive integers, and the ambient space is the reals with the usual topology. Of course, it is easy to come up with examples where a set with only isolated points has no limit point in the underlying space.

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