In general topology, a subset of a topological space is perfect if it is closed and has no isolated points.
denotes the set of all limit points of
(Some authors do not consider the empty set to be perfect.
Examples of perfect subsets of the real line
Whether a set is perfect or not (and whether it is closed or not) depends on the surrounding space.
Every topological space can be written in a unique way as the disjoint union of a perfect set and a scattered set.
[2][3] 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
These results are extended in descriptive set theory as follows: