Dense-in-itself

In general topology, a subset of a topological space is said to be dense-in-itself or crowded if has no isolated point. Equivalently, is dense-in-itself if every point of is a limit point of . Thus is dense-in-itself if and only if , where is the derived set of . A dense-in-itself closed set is called a perfect set. (In other words, a perfect set is a closed set without isolated point.) The notion of dense set is unrelated to dense-in-itself. This can sometimes be confusing, as "X is dense in X" (always true) is not the same as "X is dense-in-itself" (no isolated point). A simple example of a set that is dense-in-itself but not closed (and hence not a perfect set) is the set of irrational numbers (considered as a subset of the real 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, the set of irrationals is not closed because every rational number lies in its closure. Similarly, the set of rational numbers is also dense-in-itself but not closed in the space of real numbers. The above examples, the irrationals and the rationals, are also dense sets in their topological space, namely . As an example that is dense-in-itself but not dense in its topological space, consider . This set is not dense in but is dense-in-itself. A singleton subset of a space can never be dense-in-itself, because its unique point is isolated in it. The dense-in-itself subsets of any space are closed under unions. In a dense-in-itself space, they include all open sets. In a dense-in-itself T1 space they include all dense sets. However, spaces that are not T1 may have dense subsets that are not dense-in-itself: for example in the space with the indiscrete topology, the set is dense, but is not dense-in-itself. The closure of any dense-in-itself set is a perfect set. In general, the intersection of two dense-in-itself sets is not dense-in-itself. But the intersection of a dense-in-itself set and an open set is dense-in-itself.