Compact ultrapseudometrics and inverse systems

Abstract

A method of construction of an inverse system of finite compacta, indexed with positive reals, from a compact ultrapseudometric, is proposed. It is proved that homeomorphisms and natural preorder on the class of compact ultrapseudometrics correspond to strict isomorphisms and strict morphisms of the respective inverse systems. Properties of the poset obtained with "gluing together" of the isomorphic inverse systems, are discussed.