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Nerve complex
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[[File:Constructing nerve.png|thumb|Constructing the nerve of an [[Good_cover_(algebraic_topology)|open good cover]] containing 3 sets in the plane.]] In [[topology]], the '''nerve complex''' of a [[set family]] is an [[abstract simplicial complex|abstract complex]] that records the pattern of intersections between the sets in the family. It was introduced by [[Pavel Alexandrov]]<ref>{{cite journal |last=Aleksandroff |first=P. S. |author-link=Pavel Alexandrov |year=1928 |title=รber den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung |journal=[[Mathematische Annalen]] |volume=98 |pages=617โ635 |doi=10.1007/BF01451612 |s2cid=119590045}}</ref> and now has many variants and generalisations, among them the '''ฤech nerve''' of a cover, which in turn is generalised by [[hypercovering|hypercoverings]]. It captures many of the interesting topological properties in an algorithmic or combinatorial way.<ref>{{Cite book |last1=Eilenberg |first1=Samuel |title=Foundations of Algebraic Topology |last2=Steenrod |first2=Norman |date=1952-12-31 |publisher=[[Princeton University Press]] |isbn=978-1-4008-7749-2 |location=Princeton |doi=10.1515/9781400877492 |author1-link=Samuel Eilenberg |author2-link=Norman Steenrod}}</ref>
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