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Derived set (mathematics)
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===More properties=== Two subsets <math>S</math> and <math>T</math> are [[Separated sets|separated]] precisely when they are [[Disjoint sets|disjoint]] and each is disjoint from the other's derived set <math display=inline>S' \cap T = \varnothing = T' \cap S.</math><ref>{{harvnb|Pervin|1964|loc=p. 51}}</ref> A [[bijection]] between two topological spaces is a [[homeomorphism]] if and only if the derived set of the [[image (mathematics)|image]] (in the second space) of any subset of the first space is the image of the derived set of that subset.<ref>{{citation|first1=John G.|last1=Hocking|first2=Gail S.|last2=Young|title=Topology|year=1988|orig-date=1961|publisher=Dover|isbn=0-486-65676-4|page=[https://archive.org/details/topology00hock_0/page/4 4]|url=https://archive.org/details/topology00hock_0/page/4}}</ref> In a T<sub>1</sub> space, the derived set of any finite set is empty and furthermore, <math display=block>(S - \{p\})' = S' = (S \cup \{p\})',</math> for any subset <math>S</math> and any point <math>p</math> of the space. In other words, the derived set is not changed by adding to or removing from the given set a finite number of points.<ref>{{harvnb|Kuratowski|1966|loc=p.77}}</ref> A set <math>S</math> with <math>S \subseteq S'</math> (that is, <math>S</math> contains no [[isolated point]]s) is called [[dense-in-itself]]. A set <math>S</math> with <math>S = S'</math> is called a [[perfect set]].<ref>{{harvnb|Pervin|1964|loc=p. 62}}</ref> Equivalently, a perfect set is a closed dense-in-itself set, or, put another way, a closed set with no isolated points. Perfect sets are particularly important in applications of the [[Baire category theorem]]. 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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