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.