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In topology, a subbase (or subbasis) for a topological space with topology is a subcollection of that generates , in the sense that is the smallest topology containing . A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below. == Definition == Let be a topological space with topology . A subbase of is usually defined as a subcollection of satisfying one of the two following equivalent conditions: #The subcollection ''generates'' the topology . This means that is the smallest topology containing : any topology on containing must also contain . #The collection of open sets consisting of all finite intersections of elements of , together with the set , forms a basis for . This means that every proper open set in can be written as a union of finite intersections of elements of . Explicitly, given a point in an open set , there are finitely many sets of , such that the intersection of these sets contains and is contained in . (Note that if we use the nullary intersection convention, then there is no need to include in the second definition.) For ''any'' subcollection of the power set , there is a unique topology having as a subbase. In particular, the intersection of all topologies on containing satisfies this condition. In general, however, there is no unique subbasis for a given topology. Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the two conditions is more useful than the other. 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「Subbase」の詳細全文を読む スポンサード リンク
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