In general topology, a subset of a topological space is said to be **dense-in-itself**^{[1]}^{[2]} or **crowded**^{[3]}^{[4]}
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 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 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. For similar reasons, the set of rational numbers (also considered as a subset of the real numbers) is also dense-in-itself but not closed.

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.

- The union of any family of dense-in-itself subsets of a space X is dense-in-itself.
^{[5]} - Every open subset of a dense-in-itself space is dense-in-itself.
^{[6]} - Every dense subset of a dense-in-itself T
_{1}space is dense-in-itself.^{[7]}Note that this requires the space to be T_{1}; for example in the space with the indiscrete topology, the set is dense, but is not dense-in-itself. - A singleton subset of a space is never dense-in-itself (because its unique point is isolated in it).
- In a topological space, the closure of a dense-it-itself set is a perfect set.
^{[8]}