Namespaces
Variants
Actions

Difference between revisions of "Perfect set"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Added MSC classification, standardized bibliography and added a sentence.)
(clarify)
 
(One intermediate revision by the same user not shown)
Line 2: Line 2:
 
{{TEX|done}}
 
{{TEX|done}}
  
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]].
+
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]].
  
 
====References====
 
====References====

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=30842
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article