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In mathematics, an adherent point (also closure point or point of closure or contact point)〔 Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15.〕 of a subset ''A'' of a topological space ''X'', is a point ''x'' in ''X'' such that every open set containing ''x'' contains at least one point of . A point ''x'' is an adherent point for ''A'' if and only if ''x'' is in the closure of ''A''. This definition differs from that of a limit point, in that for a limit point it is required that every open set containing contains at least one point of ''A'' ''different from'' ''x''. Thus every limit point is an adherent point, but the converse is not true. An adherent point of ''A'' is either a limit point of ''A'' or an element of ''A'' (or both). An adherent point which is not a limit point is an isolated point. Intuitively, having an open set ''A'' defined as the area within (but not including) some boundary, the adherent points of ''A'' are those of ''A'' including the boundary. ==Examples== *If ''S'' is a subset of R which is bounded above, then sup ''S'' is adherent to ''S''. *A subset ''S'' of a metric space ''M'' contains all of its adherent points if, and only if, ''S'' is closed in ''M''. *In the interval (''a'', ''b''], ''a'' is an adherent point that is not in the interval, with usual topology of R. 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「Adherent point」の詳細全文を読む スポンサード リンク
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