Infinite-dimensional uniform polyhedra

Abstract

Uniform covers with a finite-dimensional nerve are rare (i.e., do not form a cofinal family) in many separable metric spaces of interest. To get hold on uniform homotopy properties of these spaces, a reasonably behaved notion of an infinite-dimensional metric polyhedron is needed; a specific list of desired properties was sketched by J. R. Isbell in a series of publications in 1959-64. In this paper we construct what appears to be the desired theory of uniform polyhedra; incidentally, considerable information about their metric and Lipschitz properties is obtained.