Perfect set

In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set ${\displaystyle S}$ is perfect if ${\displaystyle S=S'}$, where ${\displaystyle S'}$ denotes the set of all limit points of ${\displaystyle S}$, also known as the derived set of ${\displaystyle S}$. (Some authors do not consider the empty set to be perfect.^[1])

In a perfect set, every point can be approximated arbitrarily well by other points from the set: given any point of ${\displaystyle S}$ and any neighborhood of the point, there is another point of ${\displaystyle S}$ that lies within the neighborhood. Furthermore, any point of the space that can be so approximated by points of ${\displaystyle S}$ belongs to ${\displaystyle S}$.

Note that the term perfect space is also used, incompatibly, to refer to other properties of a topological space, such as being a G_δ space. As another possible source of confusion, also note that having the perfect set property is not the same as being a perfect set.