A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers. This set is dense-in-itself because every neighborhood of an irrational number contains at least one other irrational number . On the other hand, this set of irrationals is not closed because every rational number lies in its closure.
For similar reasons, the set of rational numbers is also dense-in-itself but not closed.
|Date of creation||2013-03-22 14:38:29|
|Last modified on||2013-03-22 14:38:29|
|Last modified by||rspuzio (6075)|