  Coarse median spaces simultaneously generalise the classes of hyperbolic
spaces and median algebras, and arise naturally in the study of the mapping
class groups and many other contexts. One issue with their definition as
originally conceived by Bowditch is the need to establish median approximations
for all finite subsets of the space, an approach which allowed the definition
of rank (a proxy for dimension) in terms of the dimensions of the approximating
spaces. Here we provide a simplification of the definition in terms of a
$4$-point condition analogous to the $4$-point condition defining
hyperbolicity. We show how to define rank in this context, and use this to give
a direct proof that rank $1$ geodesic coarse median spaces are
$\delta$-hyperbolic, bypassing Bowditch's use of asymptotic cones. A key
ingredient of the proof is a new definition of intervals in coarse median
spaces and an analysis of their interaction with geodesics.
