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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]].<ref>''Paul Alexandroff'' Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung, — Mathematische Annalen 98 (1928), стр. 617—635.</ref> | |||
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: | |||
* 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 | |||
::<math>\bigcap_{j\in J}U_j \neq \varnothing.</math> | |||
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 (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''. | |||
== Notes == | |||
<references/> | |||
== References == | |||
*Samuel Eilenberg and Norman Steenrod: ''Foundations of Algebraic Topology'', Princeton University Press, 1952, p. 234. | |||
{{DEFAULTSORT:Nerve Of A Covering}} | |||
[[Category:Topology]] | |||
[[Category:Simplicial sets]] | |||
Revision as of 13:17, 7 November 2013
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.[1]
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
- ↑ Paul Alexandroff Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung, — Mathematische Annalen 98 (1928), стр. 617—635.
References
- Samuel Eilenberg and Norman Steenrod: Foundations of Algebraic Topology, Princeton University Press, 1952, p. 234.