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| In [[mathematics]], more specifically in [[point-set topology]], the '''derived set''' of a subset ''S'' of a [[topological space]] is the set of all [[limit point]]s of ''S''. It is usually denoted by ''<math>S'</math>''.
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| The concept was first introduced by [[Georg Cantor]] in 1872 and he developed [[set theory]] in large part to study derived sets on the [[real line]].
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| == Properties ==
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| A subset ''S'' of a [[topological space]] is [[closed set|closed]] precisely when <math>S' \subseteq S</math>, when <math>S</math> contains all its limit points. Two subsets ''S'' and ''T'' are [[separated sets|separated]] precisely when they are [[disjoint sets|disjoint]] and each is disjoint from the other's derived set (though the derived sets don't need to be disjoint from each other).
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| The set ''S'' is defined to be a '''perfect set''' if <math>S = S'</math>. Equivalently, a perfect set is a closed set with no [[isolated point]]s. Perfect sets are particularly important in applications of the [[Baire category theorem]].
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| Two topological spaces are [[homeomorphic]] if and only if there is a [[bijection]] from one to the other such that the derived set of the image of any subset is the image of the derived set of that subset.
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| The '''[[Cantor–Bendixson theorem]]''' states that any [[Polish space]] can be written as the union of a countable set and a perfect set. Because any [[g-delta set|''G''<sub>δ</sub>]] subset of a Polish space is again a Polish space, the theorem also shows that any ''G''<sub>δ</sub> subset of a Polish space is the union of a countable set and a set that is perfect with respect to the [[induced topology]].
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| == Topology in terms of derived sets ==
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| Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in [[topology]]. A set of points ''X'' can be equipped with an operator <sup>*</sup> mapping subsets of ''X'' to subsets of ''X'', such that for any set ''S'' and any point ''a'':
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| # <math>\empty^* = \empty</math>
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| # <math>S^{**} \subseteq S^*</math>
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| # <math>a \in S^* \implies a \in (S \setminus \{a\})^*</math>
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| # <math>(S \cup T)^* \subseteq S^* \cup T^*</math>
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| # <math>S \subseteq T \implies S^* \subseteq T^*</math>
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| Note that given 5, 3 is equivalent to 3' below, and that 4 and 5 together are equivalent to 4' below, so we have the following equivalent axioms:
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| # <math>\empty^* = \empty</math>
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| # <math>S^{**} \subseteq S^*</math>
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| *3'. <math>S^* = (S \setminus \{a\})^*</math>
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| *4'. <math> \, (S \cup T)^* = S^* \cup T^*</math>
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| Calling a set S ''closed'' if <math>S^* \subseteq S</math> will define a topology on the space in which <sup>*</sup> is the derived set operator, that is, <math>S^* = S' \,\!</math>. If we also require that the derived set of a set consisting of a single element be empty, the resulting space will be a [[T1 space|T<sub>1</sub> space]]. In fact, 2 and 3' can fail in a space that is not T<sub>1</sub>.
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| == Cantor–Bendixson rank ==
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| For [[ordinal number]]s ''α'', the ''α''-th '''Cantor–Bendixson derivative''' of a topological space is defined by [[transfinite induction]] as follows:
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| *<math>\displaystyle X^0=X</math>
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| *<math>\displaystyle X^{\alpha+1}=(X^\alpha)'</math>
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| *<math>\displaystyle X^\lambda=\bigcap_{\alpha<\lambda}X^\alpha</math> for [[limit ordinal]]s ''λ''.
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| The transfinite sequence of Cantor–Bendixson derivatives of ''X'' must eventually be constant. The smallest ordinal ''α'' such that ''X''<sup>''α''+1</sup> = ''X''<sup>''α''</sup> is called the '''Cantor–Bendixson rank''' of ''X''.
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| == See also ==
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| * [[Perfect space]]
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| == External links ==
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| * [http://planetmath.org/encyclopedia/CantorBendixsonDerivative.html PlanetMath's article on the Cantor–Bendixson derivative]
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| == References ==
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| * {{cite book|author = Kechris, A. | title = Classical Descriptive Set Theory | edition = [[Graduate Texts in Mathematics]] 156 | publisher = Springer | year = 1995 | isbn =978-0-387-94374-9}}
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| * [[Wacław Sierpiński|Sierpiński, Wacław F.]]; translated by [[Cecilia Krieger|Krieger, C. Cecilia]] (1952). ''General Topology''. [[University of Toronto]] Press.
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| [[Category:General topology]]
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