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Difference between revisions of "Perfect set"

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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]]. 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|Cantor set]] (which is a perfect subset of $\mathbb R$) and the [[Baire space]].
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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 point]]s). In other words, $F$ coincides with its [[Derived set|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|Cantor set]] (which is a perfect subset of $\mathbb R$) and the [[Baire space]].
  
 
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Revision as of 16:48, 11 October 2014

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

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. 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=33536
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article