Simplicial approximation theorem

In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices—that is, finite simplicial complexes. The general continuous mapping between such spaces can be represented approximately by the type of mapping that is ("affine"-) linear on each simplex into another simplex, at the cost (i) of sufficient barycentric subdivision of the simplices of the domain, and (ii) replacement of the actual mapping by a homotopic one.
This theorem was first proved by L.E.J. Brouwer, by use of the Lebesgue covering theorem (a result based on compactness). It served to put the homology theory of the time—the first decade of the twentieth century—on a rigorous basis, since it showed that the topological effect (on homology groups) of continuous mappings could in a given case be expressed in a finitary way. This must be seen against the background of a realisation at the time that continuity was in general compatible with the pathological, in some other areas. This initiated, one could say, the era of combinatorial topology.
There is a further simplicial approximation theorem for homotopies, stating that a homotopy between continuous mappings can likewise be approximated by a combinatorial version.
Formal statement of the theorem.
Let formula_1 and formula_2 be two simplicial complexes. A simplicial mapping formula_3 is called a simplicial approximation of a continuous function formula_4 if for every point formula_5, formula_6 belongs to the minimal closed simplex of formula_2 containing the point formula_8. If formula_9 is a simplicial approximation to a continuous map formula_10, then the geometric realization of formula_9, formula_12 is necessarily homotopic to formula_10.
The simplicial approximation theorem states that given any continuous map formula_4 there exists a natural number formula_15 such that for all formula_16 there exists a simplicial approximation formula_17 to formula_10 (where formula_19 denotes the barycentric subdivision of formula_1, and formula_21 denotes the result of applying barycentric subdivision formula_22 times.)