This paper investigates the
structure of hyper-real z-ultrafilters on completely regular, Hausdorff spaces in an
attempt to describe their structure in manageable terms. A consequence of this
investigation is a scheme for classifying these z-filters based on the complexity of
their structure. It is shown that the real numbers with the usual topology exhibit
hyper-real z-ultrafilters within each category of the classification. The paper closes
with a discussion of how the action of z-filters in one category influence
those in the other categories with particular applications to the study of
C♯(X).
