of a topological space is said to be dense-in-itself[1][2] or crowded[3][4] if
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).
This set is dense-in-itself because every neighborhood of an 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
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.
[5] In a dense-in-itself space, they include all open sets.
[6] In a dense-in-itself T1 space they include all dense sets.
This article incorporates material from Dense in-itself on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.