In mathematics, a subset A {\displaystyle A} of a topological space is said to be dense-in-itself if A {\displaystyle A} contains no isolated points. Every dense-in-itself closed set is perfect. Conversely, every perfect set is dense-in-itself. 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 (considered as a subset of the real numbers). This set is dense-in-itself because every neighborhood of an irrational number ...

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A; that is, the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, a subset A of a topological space X is dense in...