  Current techniques for generating a knowledge space, such as QUERY,
guarantees that the resulting structure is closed under union, but not that it
satisfies wellgradedness, which is one of the defining conditions for a
learning space. We give necessary and sufficient conditions on the base of a
union-closed set family that ensures that the family is well-graded. We
consider two cases, depending on whether or not the family contains the empty
set. We also provide algorithms for efficiently testing these conditions, and
for augmenting a set family in a minimal way to one that satisfies these
conditions.
