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| − | The collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d0313101.png" /> of all limit points of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d0313102.png" /> in a topological space (cf. [[Limit point of a set|Limit point of a set]]). A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d0313103.png" /> that coincides with its derived set is called perfect. | + | The collection $M'$ of all limit points of a set $M$ in a topological space (cf. [[Limit point of a set]]). A set $M$ that is contained in its derived set is called [[dense-in-itself set|dense-in-itself]]; if in addition $M$ is closed, it is termed a [[perfect set]]. |
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| | This process can be iterated. | | This process can be iterated. |
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| − | In general one defines, for an [[Ordinal number|ordinal number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d0313104.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d0313106.png" />-th derived set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d0313107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d0313108.png" />, as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d0313109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131010.png" /> is the derived set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131011.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131012.png" /> is a limit ordinal then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131013.png" />. | + | In general one defines, for an [[Ordinal number|ordinal number]] $\alpha$, the $\alpha$-th derived set of $X$, $X^{(\alpha)}$, as follows: $X^{(0)} = X$, $X^{(\alpha+1)}$ is the derived set of $X^{(\alpha)}$, and if $\lambda$ is a limit ordinal then $X^{(\lambda)} = \bigcap_{\alpha < \lambda} X^{(\alpha)}$. |
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| − | One then shows that there is a first ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131014.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131015.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131017.png" /> is called scattered; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131019.png" /> is called the perfect kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131020.png" />. | + | One then shows that there is a first ordinal number $\alpha = \alpha_X$ such that $X^{(\alpha+1)} = X^{(\alpha)}$. If $X^{(\alpha)} = \emptyset$, then $X$ is called scattered; if $X^{(\alpha)} \neq \emptyset$, then $X^{(\alpha)}$ is called the perfect kernel of $X$. |
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| − | In this way one can prove the Cantor–Bendixson theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131021.png" /> is a subspace of the real line, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131022.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131023.png" /> a countable set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131024.png" /> a perfect set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131025.png" />. | + | In this way one can prove the Cantor–Bendixson theorem: If $X$ is a subspace of the real line, then $X = C \cup P$, with $C$ a countable set, $P$ a perfect set and $C \cap P = \emptyset$. |
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| − | For this reason <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131026.png" /> is sometimes called the Cantor–Bendixson height of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031310/d03131027.png" />. Perfect spaces are sometimes called dense-in-itself. | + | For this reason $\alpha_X$ is sometimes called the Cantor–Bendixson height of $X$. |
Latest revision as of 20:29, 13 December 2017
The collection $M'$ of all limit points of a set $M$ in a topological space (cf. Limit point of a set). A set $M$ that is contained in its derived set is called dense-in-itself; if in addition $M$ is closed, it is termed a perfect set.
This process can be iterated.
In general one defines, for an ordinal number $\alpha$, the $\alpha$-th derived set of $X$, $X^{(\alpha)}$, as follows: $X^{(0)} = X$, $X^{(\alpha+1)}$ is the derived set of $X^{(\alpha)}$, and if $\lambda$ is a limit ordinal then $X^{(\lambda)} = \bigcap_{\alpha < \lambda} X^{(\alpha)}$.
One then shows that there is a first ordinal number $\alpha = \alpha_X$ such that $X^{(\alpha+1)} = X^{(\alpha)}$. If $X^{(\alpha)} = \emptyset$, then $X$ is called scattered; if $X^{(\alpha)} \neq \emptyset$, then $X^{(\alpha)}$ is called the perfect kernel of $X$.
In this way one can prove the Cantor–Bendixson theorem: If $X$ is a subspace of the real line, then $X = C \cup P$, with $C$ a countable set, $P$ a perfect set and $C \cap P = \emptyset$.
For this reason $\alpha_X$ is sometimes called the Cantor–Bendixson height of $X$.
How to Cite This Entry:
Derived set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derived_set&oldid=11226
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article