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In mathematics, the zero set of a real-valued function ''f'' : ''X'' → R (or more generally, a function taking values in some additive group) is the subset of ''X'' (the inverse image of ). In other words, the zero set of the function ''f'' is the subset of ''X'' on which . The cozero set of ''f'' is the complement of the zero set of ''f'' (i.e. the subset of ''X'' on which ''f'' is nonzero). Zero sets are important in several branches of geometry and topology. ==Topology== In topology, zero sets are defined with respect to continuous functions. Let ''X'' be a topological space, and let ''A'' be a subset of ''X''. Then ''A'' is a zero set in ''X'' if there exists a continuous function ''f'' : ''X'' → R such that : A cozero set in ''X'' is a subset whose complement is a zero set. Every zero set is a closed set and a cozero set is an open set, but the converses does not always hold. In fact: *A topological space ''X'' is completely regular if and only if every closed set is the intersection of a family of zero sets in ''X''. Equivalently, ''X'' is completely regular if and only if the cozero sets form a basis for ''X''. *A topological space is perfectly normal if and only if every closed set is a zero set (equivalently, every open set is a cozero set). 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「Zero set」の詳細全文を読む スポンサード リンク
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