  For a countable discrete group {\Gamma} and a minimal {\Gamma}-space X, we
study the notion of ({\Gamma}, X)-boundary, which is a natural generalization
of the notion of topological {\Gamma}-boundary in the sense of Furstenberg. We
give characterizations of the ({\Gamma}, X)-boundary in terms of essential or
proximal extensions. The characterization is used to answer a problem of Hadwin
and Paulsen in negative. As an application, we find necessary and sufficient
condition for the corresponding reduced crossed product to be exact.
