Perfect set

Examples of perfect subsets of the real line ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle S=[0,1]\cap \mathbb {Q} }$ is perfect as a subset of the space ${\displaystyle \mathbb {Q} }$ but not perfect as a subset of the space ${\displaystyle \mathbb {R} }$.

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.

Cantor also showed that every non-empty perfect subset of the real line has cardinality ${\displaystyle 2^{\aleph _{0}}}$, 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 ${\displaystyle 2^{\aleph _{0}}}$. If X is a separable, complete metric space with no isolated points, the cardinality of X is exactly ${\displaystyle 2^{\aleph _{0}}}$.
• 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 ${\displaystyle 2^{\aleph _{0}}}$.

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

• Engelking, Ryszard, General Topology, Heldermann Verlag Berlin, 1989. ISBN 3-88538-006-4
• Kechris, A. S. (1995), Classical Descriptive Set Theory, Berlin, New York: Springer-Verlag, ISBN 3540943749
• Levy, A. (1979), Basic Set Theory, Berlin, New York: Springer-Verlag
• Pearl, Elliott, ed. (2007), Open problems in topology. II, Elsevier, ISBN 978-0-444-52208-5, MR 2367385