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Difference between revisions of "Perfect set"

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A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072110/p0721101.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072110/p0721102.png" /> which is both closed and dense-in-itself (that is, has no isolated points). In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072110/p0721103.png" /> coincides with its [[Derived set|derived set]]. Examples are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072110/p0721104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072110/p0721105.png" /> 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 (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]].
 
 
====Comments====
 
  
  
 
====References====
 
====References====
 
<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