N-skeleton 2977910 195258038 2008-03-02T04:05:04Z Mct mht 1190575 rm {{technical}}. + one statement in intro. {{lowercase}} :''This article is not about the [[topological skeleton]] concept of [[computer graphics]]'' In [[mathematics]], particularly in [[algebraic topology]], the '''''n''-skeleton''' of a [[topological space]] ''X'' presented as a [[simplicial complex]] (resp. [[CW complex]]) refers to the [[subspace]] ''X''<sub>''n''</sub> that is the union of the simplices of ''X'' (resp. cells of ''X'') of dimensions ''m'' &le; ''n''. In other words, given an inductive definition of a complex, the ''n''-skeleton is obtained by stopping at the ''n''-th step. These subspaces increase with ''n''. The 0-skeleton is a [[discrete space]], and the 1-skeleton a [[topological graph]]. The skeletons of a space are used in [[obstruction theory]], to construct [[spectral sequence]]s by means of [[filtration]]s, and generally to make [[Mathematical induction|inductive argument]]s. They are particularly important when ''X'' has infinite dimension, in the sense that the ''X''<sub>''n''</sub> do not become constant as ''n'' &rarr; &infin;. == In geometry == In [[geometry]], a ''k''-skeleton of ''n''-[[polytope]] P (functionally represented as skel<sub>''k''</sub>(''P'')) consists of all ''i''-polytope elements of dimension up to ''k''.<ref>Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0 (Page 29)</ref> == References == {{reflist}} == External links == * [http://mathworld.wolfram.com/Skeleton.html Mathworld: Skeleton] {{topology-stub}} [[Category:Algebraic topology]] [[Category:General topology]]