A note on b_f -spaces and on the distribution of the functor of the Dieudonné completion

Abstract

A subset B of a space X is said to be bounded (in X) if the restriction to B of every real-valued continuous function on X is bounded. A real-valued function on X is called bf -continuous if its restriction to each bounded subset of X has a continuous extension to the whole space X. bf -spaces are spaces such that bf -continuous functions are continuous. We take advantage to the exponential map in the realm of bf -spaces in order to study bf -extensions of bf -continuous functions. This allows us to improve several results concerning the distribution of the functor of the Dieudonné completion. We also prove that a relative version of the classical Glicksberg’s theorem characterizing the product of two pseudocompact spaces is valid for kr- spaces. In the last section we show that bf -hemibounded groups are Moscow spaces and, consequently, they are strong-PT-groups.