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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''.
| | Hi there! :) My name is Annabelle, I'm a student studying Japanese Studies from Saint-Quentin, France.<br><br>Also visit my web site - [http://ebook-pdfree.blogspot.com/2014/08/free-download-orange-is-new-black-pdf.html Orange Is the New Black pdf] |
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| The notion of nerve was introduced by [[Pavel Alexandrov]].<ref>''Paul Alexandroff'' Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung, — Mathematische Annalen 98 (1928), стр. 617—635.</ref>
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| Given an [[index set]] ''I'', and [[open set]]s ''U<sub>i''</sub> contained in ''X'', the ''nerve'' ''N'' is the set of finite subsets of ''I'' defined as follows:
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| * a finite set ''J'' ⊆ ''I'' belongs to ''N'' if and only if the intersection of the ''U<sub>i</sub>'' whose subindices are in ''J'' is non-empty. That is, if and only if
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| ::<math>\bigcap_{j\in J}U_j \neq \varnothing.</math>
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| Obviously, if ''J'' belongs to ''N'', then any of its subsets is also in ''N''. Therefore ''N'' is an abstract simplicial complex.
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| 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 (set theory)|intersection]] indexed by a set in ''N'' is also [[contractible space|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''.
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| == Notes ==
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| <references/>
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| == References ==
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| *Samuel Eilenberg and Norman Steenrod: ''Foundations of Algebraic Topology'', Princeton University Press, 1952, p. 234.
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| {{DEFAULTSORT:Nerve Of A Covering}}
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| [[Category:Topology]]
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| [[Category:Simplicial sets]]
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Hi there! :) My name is Annabelle, I'm a student studying Japanese Studies from Saint-Quentin, France.
Also visit my web site - Orange Is the New Black pdf