Tolerance space theory and some applications (Q1073397)

This paper is a review of tolerance space theory. It consists of three parts titled, ''Philosophy'', ''Techniques'', and ''Application''. Part I introduces the notion of tolerance spaces and the notion of toleomorphism. This latter is a tolerance on the class of tolerance spaces. Part II discusses the homotopy; homology, and dimension theory of tolerance spaces. A particularly striking result is that, in the tolerance space sense, all manifolds are 0, 1, 2, or 3 dimensional. This is a consequence of the existence of an embedding of the 3-cube in the 4- cube which is \(\epsilon\)-dense and preserves \(2\epsilon\)-small arcs. Part III describes applications to the Dirichlet problem, information transmission on continuous channels, and biology. There is a concluding section on questions for further research.