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In mathematics, the nerve of an open covering is a construction in topology, of an abstract simplicial complex from an open covering of a topological space ''X''. The notion of nerve was introduced by Pavel Alexandrov.〔''Paul Alexandroff'' Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung, — Mathematische Annalen 98 (1928), стр. 617—635.〕 Given an index set ''I'', and open sets ''Ui'' contained in ''X'', the ''nerve'' ''N'' is the set of finite subsets of ''I'' defined as follows: * a finite set ''J'' ⊆ ''I'' belongs to ''N'' if and only if the intersection of the ''Ui'' whose subindices are in ''J'' is non-empty. That is, if and only if :: Obviously, if ''J'' belongs to ''N'', then any of its subsets is also in ''N''. Therefore ''N'' is an abstract simplicial complex. In general, the complex ''N'' need not reflect the topology of ''X'' accurately. For example we can cover any ''n''-sphere with two contractible sets ''U'' and ''V'', in such a way that ''N'' is a 1-simplex. However, if we also insist that the open sets corresponding to every intersection indexed by a set in ''N'' is also contractible, the situation changes. This means for instance that a circle covered by three open arcs, intersecting in pairs in one arc, is modelled by a homeomorphic complex, the geometrical realization of ''N''. == Notes == 〔 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「nerve of a covering」の詳細全文を読む スポンサード リンク
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