In general topology, a subset ${\displaystyle A}$ of a topological space is said to be dense-in-itself^[1][2] or crowded^[3][4] if ${\displaystyle A}$ has no isolated point. Equivalently, ${\displaystyle A}$ is dense-in-itself if every point of ${\displaystyle A}$ is a limit point of ${\displaystyle A}$. Thus ${\displaystyle A}$ is dense-in-itself if and only if ${\displaystyle A\subseteq A'}$, where ${\displaystyle A'}$ is the derived set of ${\displaystyle A}$.

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).

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

A singleton subset of a space ${\displaystyle X}$ 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.^[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.^[7] However, spaces that are not T₁ may have dense subsets that are not dense-in-itself: for example in the dense-in-itself space ${\displaystyle X=\{a,b\}}$ with the indiscrete topology, the set ${\displaystyle A=\{a\}}$ is dense, but is not dense-in-itself.

The closure of any dense-in-itself set is a perfect set.^[8]

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.

• Nowhere dense set
• Glossary of topology
• Dense order