Uniformly open mappings and uniform embeddings of function spaces

We study the class of uniformly open mappings between uniform spaces. Furthermore the Hausdorff uniformity on the space Cb(X) of bounded continuous real-valued functions of a supercomplete uniform space X is investigated. With the help of our results we prove that for the Menger curve μ1 the space Chb(μ1) is a universal uniform coretract for the spaces Chb(X), where X is an arbitrary Peano continuum and Chb(X) denotes the completion of CBb(X).