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 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 2ω 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 the Cantor space to
X, and so
X has cardinality at least 2^{\aleph_0}. ==See also==