# Perfect set

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2010 Mathematics Subject Classification: Primary: 54A05 [MSN][ZBL]

A subset \$F\$ of a topological space \$X\$ which is both closed and dense-in-itself. In other words, \$F\$ coincides with its derived set. A perfect topological space is therefore a topological space with no isolated points. Examples are \$\mathbb R^n\$ and \$\mathbb C^n\$ (with the standard topology induced by the Euclidean distance), the Cantor set (which is a perfect subset of \$\mathbb R\$) and the Baire space.

#### References

 [Ar] 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://encyclopediaofmath.org/index.php?title=Perfect_set&oldid=42513
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article