Perfect set explained

In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set

S is perfect if S = S' , where S' denotes the set of all limit points of S , also known as the derived set of S .

In a perfect set, every point can be approximated arbitrarily well by other points from the set: given any point of

S and any neighborhood of the point, there is another point of S that lies within the neighborhood. Furthermore, any point of the space that can be so approximated by points of S belongs to 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.

Examples

R 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.

Whether a set is perfect or not (and whether it is closed or not) depends on the surrounding space. For instance, the set

S = [ 0 , 1 ]\cap \Q is perfect as a subset of the space \Q but not perfect as a subset of the space R , 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.^{[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.

2 ^{\aleph _{0}} , 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^ω can be continuously embedded into X. Thus X has cardinality at least

2 ^{\aleph _{0}} . If X is a separable, complete metric space with no isolated points, the cardinality of X is exactly 2 ^{\aleph _{0}} .

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

2 ^{\aleph _{0}} .

See also

• Dense-in-itself
• Finite intersection property
• Subspace topology

References

• Book: Engelking . Ryszard . Ryszard Engelking . General Topology . Heldermann Verlag . Berlin . 3-88538-006-4 . 1989 .

Notes and References

1. Engelking, problem 1.7.10, p. 59
2. Web site: Uniqueness of decomposition into perfect set and scattered set - Mathematics Stack Exchange.