  It is well-known that point-set topology (without additional structure) lacks
the capacity to generalize the analytic concepts of completeness, boundedness,
and other typically-metric properties. The ability of metric spaces to capture
this information is tied to the fact that the topology is generated by open
balls whose radii can be compared. In this paper, we construct spaces that
generalize this property, called $\beta$-spaces, and show that they provide a
framework for natural definitions of the above concepts. We show that
$\beta$-spaces are strictly more general than uniform spaces, a common
generalization of metric spaces. We then conclude by proving generalizations of
several typically-metric theorems, culminating in a broader statement of the
Contraction Mapping Theorem.
