# Perfect set

In mathematics, in the field of topology, a perfect set is a subset of a topological space that is closed and has no isolated points. Equivalently: the set ${\displaystyle S}$ is perfect if ${\displaystyle S=S'}$, where ${\displaystyle S'}$ denotes the set of all limit points of ${\displaystyle S}$, also known as the derived set of ${\displaystyle S}$.

In a perfect set, every point can be approximated arbitrarily well by other points from the set: given any point of ${\displaystyle S}$ and any topological neighborhood of the point, there is another point of ${\displaystyle S}$ that lies within the neighborhood.

Note that the term perfect space is also used, incompatibly, to refer to other properties of a topological space, such as being a Gδ space.

## Examples

Examples of perfect subsets of the real line ${\displaystyle \mathbb {R} }$ are: the empty set, all closed intervals and the Cantor set. The latter is noteworthy in that it is totally disconnected.

## Connection with other topological properties

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 2ω 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 ${\displaystyle 2^{\aleph _{0}}}$.