Perfect set

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In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set is perfect if , where denotes the set of all limit points of , also known as the derived set of .

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In a perfect set, every point can be approximated arbitrarily well by other points from the set: given any point of and any neighborhood of the point, there is another point of that lies within the neighborhood. Furthermore, any point of the space that can be so approximated by points of belongs to .

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 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.

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 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 , the cardinality of the continuum. These results are extended in descriptive set theory as follows:

See also

Notes

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