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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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A subset $F$ of a topological space $X$ which is both closed and [[dense-in-itself set|dense-in-itself]]. In other words, $F$ coincides with its [[derived set]]. A perfect topological space is therefore a topological space with no [[isolated point]]s. 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]].
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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>
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|valign="top"|{{Ref|Ar}}|| A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  pp. 62, 1442ff  (Translated from Russian)
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Latest revision as of 20:27, 13 December 2017

2020 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=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
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