Dense-in-itself

In general topology, a subset $$A$$ of a topological space is said to be dense-in-itself or crowded if $$A$$ has no isolated point. Equivalently, $$A$$ is dense-in-itself if every point of $$A$$ is a limit point of $$A$$. Thus $$A$$ is dense-in-itself if and only if $$A\subseteq A'$$, where $$A'$$ is the derived set of $$A$$.

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 distinct from 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).

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 $$x$$ contains at least one other irrational number $$y \neq x$$. 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 $$\mathbb{R}$$. As an example that is dense-in-itself but not dense in its topological space, consider $$\mathbb{Q} \cap [0,1]$$. This set is not dense in $$\mathbb{R}$$ but is dense-in-itself.

Properties
A singleton subset of a space $$X$$ 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 dense-in-itself space $$X=\{a,b\}$$ with the indiscrete topology, the set $$A=\{a\}$$ 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.