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In mathematics, the upper topology on a partially ordered set ''X'' is the coarsest topology in which the closure of a singleton is the order section for each . If is a partial order, the upper topology is the least order consistent topology in which the open sets are the up-sets. The lower topology induced by the preorder is defined similarly in terms of the down-sets. The preoder inducing the upper topology is its specialization preorder, but the specialization preorder of the lower topology is opposite to the inducing preorder. The real upper topology is most naturally defined on the upper-extended real line by the system of open sets. Similarly, the real lower topology . ==References== * * * 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「upper topology」の詳細全文を読む スポンサード リンク
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