The $k$ notation is generally used to describe Friedmann Robertson Walker cosmological models. These are built on the assumptions of homogeneity and isotropy. The spacetime can be described as being foliated by spatial slices of constant curvature. The k value is the sign of this spatial curvature if the {-1, 0, +1} convention is adopted. As the curvature is a constant, it makes sense to talk of its sign. Further details here.
This post imported from StackExchange Physics at 2014-03-22 16:57 (UCT), posted by SE-user twistor59