Perfect set

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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.


[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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Perfect set. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article