Perfect set

In general topology, a subset of a topological space is perfect if it 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

${\displaystyle S}$

${\displaystyle S}$

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

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

is not the same as being a perfect set.

Examples

Examples of perfect subsets of the

real line

${\displaystyle \mathbb {R} }$ are the

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 ${\displaystyle \mathbb {R} }$.

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 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

Notes

References

• ISBN 3-88538-006-4
  .
• ISBN 3540943749
• Springer-Verlag
• Pearl, Elliott, ed. (2007), Open problems in topology. II,
  MR 2367385