In general topology, a subset of a topological space is said to be dense-in-itselfSteen & Seebach, p. 6Engelking, p. 25 or crowded if has no isolated point of . 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 property of being dense-in-itself depends only on the subspace topology of , namely, is a dense-in-itself subset of if and only if is a dense-in-itself subset of . Some authors would say that is a perfect space in its subspace topology (note that the phrase perfect space is ambiguous).
The notion of dense set is distinct from dense-in-itself. This can sometimes be confusing, as " is dense in " (always true) is not the same as " is dense-in-itself" (contains no isolated points).
Examples 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.
Properties A singleton is never dense-in-itself.
The dense-in-itself subsets of any space are closed under arbitrary unions.Engelking, 1.7.10, p. 59 In a dense-in-itself space, they include all open sets.Kuratowski, p. 78 In a dense-in-itself T1 space they include all dense sets.Kuratowski, p. 78 However, spaces that are not T1 may have dense subsets that are not dense-in-itself: for example in the dense-in-itself 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.Kuratowski, p. 77
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.
See also Nowhere dense set Glossary of topology Dense order
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