Perfect set

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.

Every topological space can be written in a unique way as the disjoint union of a perfect set and a scattered set.[1][2]

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.