Difference between revisions of "Perfect set"
From Encyclopedia of Mathematics
m (TeX encoding is done) |
(Added MSC classification, standardized bibliography and added a sentence.) |
||
| Line 1: | Line 1: | ||
| + | {{MSC|54A05}} | ||
{{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]]. Examples are $\mathbb R^n$ | + | 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]]. |
| − | |||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |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) | ||
| + | |- | ||
| + | |} | ||
Revision as of 11:30, 2 December 2013
2020 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=29212
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