Page "Barycentric subdivision" Paragraph 12

Barycentric subdivision is an important tool in simplicial homology theory, where it is used as a means of obtaining finer simplicial complexes ( containing the original ones, i. e. with more simplices ).

This in turn is crucial to the simplicial approximation theorem, which roughly states that one can approximate any continuous function between polyhedra by a ( finite ) simplicial map, given a sufficient amount of subdivision of the respective simplicial complexes whom they realize.

Ultimately, this approximation technique is a standard ingredient in the proof that simplicial homology groups are topological invariants.