# n-skeleton

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*

_{n}that is the union of the simplices of

*X*(resp. cells of

*X*) of dimensions

*m*≤

*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 sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when *X* has infinite dimension, in the sense that the *X*_{n} do not become constant as *n* → ∞.

In geometry, a *k*-skeleton of *n*-polytope P (functionally represented as skel_{k}(*P*)) consists of all *i*-polytope elements of dimension up to *k*.^{[1]}

(The boundary and degeneracy morphisms are given by various projections and diagonal embeddings, respectively.)

The above constructions work for more general categories (instead of sets) as well, provided that the category has fiber products. The coskeleton is needed to define the concept of hypercovering in homotopical algebra and algebraic geometry.^{[3]}