In mathematics, a subset A of a topological space's said to be dense-in-itself if A contains no isolated points. Every dense-in-itself closed set's perfect. Conversely, every perfect set's dense-in-itself. A simple example of a set which's dense-in-itself but not closed (and hence not a perfect set)'s the subset of irrational numbers. This set's dense-in-itself because every neighborhood of an irrational number x contains at least one other irrational number y eq x. On the other hand, this set o… (More on Dense-in-itself)