I was glanced at this question here and it cause me to wonder the following:
Question: Is there a simple description of the subsets $L \subset [0,1]$ with the property that there exists a sequence in $[0,1] \setminus L$ whose limit points are exactly $L$?
A few things are pretty easy to see
- Any finite nonempty $L$ has this property.
- Some strange sets have this property. For instance, I think you can get the Cantor set from, say, an enumeration of the midpoints of the intervals deleted during the "take out the middle thirds" construction.
Anyway, I thought it might be a fun problem to find a precise description.