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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 points of ''S''. It is usually denoted by ''''. 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. == Properties == A subset ''S'' of a topological space is closed precisely when , i.e. when contains all its limit points. Two subsets ''S'' and ''T'' are separated precisely when they are disjoint and each is disjoint from the other's derived set (though the derived sets don't need to be disjoint from each other). The set ''S'' is defined to be a perfect set if . Equivalently, a perfect set is a closed set with no isolated points. Perfect sets are particularly important in applications of the Baire category theorem. 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. 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''δ subset of a Polish space is again a Polish space, the theorem also shows that any ''G''δ subset of a Polish space is the union of a countable set and a set that is perfect with respect to the induced topology. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Derived set (mathematics)」の詳細全文を読む スポンサード リンク
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