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 formula_1 is perfect if formula_2, where formula_3 denotes the set of all limit points of formula_1, also known as the derived set of formula_1.
In a perfect set, every point can be approximated arbitrarily well by other points from the set: given any point of formula_1 and any neighborhood of the point, there is another point of formula_1 that lies within the neighborhood. Furthermore, any point of the space that can be so approximated by points of formula_1 belongs to formula_1.
Note that the term "perfect space" is also used, incompatibly, to refer to other properties of a topological space, such as being a Gδ space.
Examples.
Examples of perfect subsets of the real line formula_10 are: the empty set, all closed intervals, the real line itself, and the Cantor set. The latter is noteworthy in that it is totally disconnected.
Connection with other topological properties.
Every topological space can be written in a unique way as the disjoint union of a perfect set and a scattered set.
Cantor proved that every closed subset of the real line can be uniquely written as the disjoint union of a perfect set and a countable set. This is also true more generally for all closed subsets of Polish spaces, in which case the theorem is known as the Cantor–Bendixson theorem.
Cantor also showed that every non-empty perfect subset of the real line has cardinality formula_11, the cardinality of the continuum. These results are extended in descriptive set theory as follows: