How do I say a set is nowhere discrete?

Question

How do I say a set is nowhere discrete?

If the vicinity of $x$ is $R(x,\epsilon)=(x-\epsilon,x+\epsilon)$

then $X$ is nowhere discrete iff $x\subset R(x,\epsilon)\cap x \quad\forall x\in X\quad\forall\epsilon>0$

Not sure if that's quite right. I'm trying to say no matter how small the vicinity, there's always another number in there. It seems to work but I'm sure there must be a better way.

Answer 1

You can say $X$ has no isolated points, or that every point of $X$ is a limit point.

You can say $X$ is dense in itself, though that terminology is not as common.

If $X$ is also closed, then you could say $X$ is a perfect set.