of points. It can also be shown that in a T1 space, for any subset A set with is called dense-in-itself and can contain no isolated points. A set with is called a perfect set. Equivalently, a perfect set is a closed dense-in-itself set, or, put another way, a closed set with no isolated points. Perfect sets are particularly important in applications of the Baire category theorem. The Cantor–Bendixson theorem states that any Polish space can be written as the union of a countable set and a perfect set. Because any Gδ subset of a Polish space is again a Polish space, the theorem also shows that any Gδ subset of a Polish space is the union of a countable set and a set that is perfect with respect to the induced topology. Topology in terms of derived sets Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in topology. A set of points can be equipped with an operator mapping subsets of to subsets of such that for any set and any point : implies implies Calling a set if will define a topology on the space in which is the derived set operator, that is, Cantor–Bendixson rank For ordinal numbers the -th of a topological space is defined by repeatedly applying the derived set operation using transfinite recursion as follows: for limit ordinals The transfinite sequence of Cantor–Bendixson derivatives of must eventually be constant. The smallest ordinal such that is called the of This investigations into the derivation process was one of the motivations for introducing ordinal numbers by Georg Cantor. See also Notes Proofs References Further reading Sierpiński, Wacław F.; translated by Krieger, C. Cecilia (1952). General Topology. University of Toronto Press. External links PlanetMath's article on the Cantor–Bendixson derivative General topology