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, also known as the derived set. (Some authors do not consider the empty set to be perfect.)

and any neighborhood.

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

Examples

Examples of perfect subsets of the real line 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.

, since it fails to be closed in the latter.

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, the cardinality of the continuum. These results are extended in descriptive set theory as follows:

• If X is a complete metric space with no isolated points, then the Cantor space 2 can be continuously embedded into X. Thus, X. If X is a separable, complete metric space with no isolated points, the cardinality of X.

• If X is a locally compact Hausdorff space with no isolated points, there is an injective function (not necessarily continuous) from the Cantor space to X, and so X.

See also

• Dense-in-itself

• Finite intersection property

• Subspace topology

Notes

References