# Difference between revisions of "Perfect set"

From Encyclopedia of Mathematics

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− | + | A subset $F$ of a topological space $X$ which is both closed and dense-in-itself (that is, has no isolated points). In other words, $F$ coincides with its [[Derived set|derived set]]. Examples are $\mathbb R^n$, $\mathbb C^n$ and the [[Cantor set|Cantor set]]. | |

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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 62, 1442ff (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 62, 1442ff (Translated from Russian)</TD></TR></table> |

## Revision as of 15:53, 15 December 2012

A subset $F$ of a topological space $X$ which is both closed and dense-in-itself (that is, has no isolated points). In other words, $F$ coincides with its derived set. Examples are $\mathbb R^n$, $\mathbb C^n$ and the Cantor set.

#### References

[a1] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 62, 1442ff (Translated from Russian) |

**How to Cite This Entry:**

Perfect set.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Perfect_set&oldid=29212

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article