  The relationship between the operator approximation property and the strong
operator approximation property has deep significance in the theory of operator
algebras. The original definitions of Effros and Ruan, unlike the classical
analogues, make no mention of compact operators or compact sets. In this paper
we introduce ``compact matrix sets'' which correspond to the two different
operator approximation properties, and show that a space has the operator
approximation property if and only if the ``operator compact'' operators are
contained in the closure of the finite rank operators. We also investigate when
the two types of compactness agree, and introduce a natural condition which
guarantees that they do.
