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In general topology and related areas of mathematics, the initial topology (or weak topology or limit topology or projective topology) on a set , with respect to a family of functions on , is the coarsest topology on ''X'' that makes those functions continuous. The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these. The dual construction is called the final topology. ==Definition== Given a set ''X'' and an indexed family (''Y''''i'')''i''∈''I'' of topological spaces with functions : the initial topology τ on is the coarsest topology on ''X'' such that each : is continuous. Explicitly, the initial topology may be described as the topology generated by sets of the form , where is an open set in . The sets are often called cylinder sets. If ''I'' contains just one element, all the open sets of are cylinder sets. 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「initial topology」の詳細全文を読む スポンサード リンク
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