Characterization of properties of spaces in terms of properties of their continuous images (Q1074164)

A real function f on space X is strictly b-continuous if the contraction of f onto each bounded set in X coincides with the contraction onto the same set of some function that is continuous on X. Space X is called a \(b_ f\)-space if every strictly b-continuous function on X is continuous. Obviously, the class of \(b_ f\)-spaces contains all pseudocompact spaces, all k-spaces, and all quasi-k-spaces in the sense of Nagata. The main result gives the following characterization of pseudocompactness: A Tikhonov space is pseudocompact if and only if every Tikhonov space that is the continuous image of X is a \(b_ f\)-space.