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What do you call a set of points with the following property?

For any point and any number $\epsilon$, you can find another point in the set that is less than $\epsilon$ away from the first point.

An example would be the rationals, because for any $\epsilon$ there is some positive rational number smaller than it, and you can just add that number to your point to get the required second point.

Thanks!

badatmath
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    Such a set is dense-in-itself, or has no isolated points. Such sets are also sometimes called perfect sets, but I prefer to avoid this terminology, as perfect has other meanings. – Brian M. Scott May 17 '12 at 08:04

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Turning my comment into an answer:

Such a set is said to be dense-in-itself. The term perfect is also sometimes used, but I prefer to avoid it, since it has other meanings in general topology. One can also describe such a set by saying that it has no isolated points. All of this terminology applies to topological spaces in general, not just to metric spaces.

Brian M. Scott
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