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Simplicial approximation theorem
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the simplicial approximation theorem is a foundational result for <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, guaranteeing that <a href="/facts/Continuous_mapping/sKbl02pB">continuous mappings</a> can be (by a slight deformation) approximated by ones that are <a href="/facts/Piecewise/nlQwptpe">piecewise</a> of the simplest kind. It applies to mappings between spaces that are built up from <a href="/facts/Simplex/0FCfNRN8">simplices</a>—that is, finite <a href="/facts/Simplicial_complex/fF2282CF">simplicial complexes</a>. The general continuous mapping between such spaces can be represented approximately by the type of mapping that is (<i>affine</i>-) linear on each simplex into another simplex, at the cost (i) of sufficient <a href="/facts/Barycentric_subdivision/hsk1bw6b">barycentric subdivision</a> of the simplices of the domain, and (ii) replacement of the actual mapping by a <a href="/facts/Homotopic/1nWrPxdj">homotopic</a> one. </p><p>This theorem was first proved by <a href="/facts/L.E.J._Brouwer/6q6ULf3b">L.E.J. Brouwer</a>, by use of the <a href="/facts/Lebesgue_covering_theorem/xdCJLVdw">Lebesgue covering theorem</a> (a result based on <a href="/facts/Compactness/d0cgXJH7">compactness</a>). It served to put the <a href="/facts/Homology_theory/VFJmryEW">homology theory</a> of the time—the first decade of the twentieth century—on a rigorous basis, since it showed that the topological effect (on <a href="/facts/Homology_group/VFJmryEW">homology groups</a>) of continuous mappings could in a given case be expressed in a <a href="/facts/Finitary/B0eHCDjw">finitary</a> way. This must be seen against the background of a realisation at the time that continuity was in general compatible with the <a href="/facts/Pathological_(mathematics)/KmKHjJ1H">pathological</a>, in some other areas. This initiated, one could say, the era of <a href="/facts/Combinatorial_topology/NmwG5Uho">combinatorial topology</a>. </p><p>There is a further simplicial approximation theorem for homotopies, stating that a <a href="/facts/Homotopy/1nWrPxdj">homotopy</a> between continuous mappings can likewise be approximated by a combinatorial version. </p>