Let M be a finitely generated module over a semi-local ring R with Jacobson radical J. If M is complete in the J-adic metric then, given a decreasing sequenceof nonempty closed subsets of M with intersection C0and a natural number n, necessarily CiC0+JnM for large i. This property is equivalent to the completeness of M in the J-adic metric, as well as to the completeness in the HausdorfF metric of the space of nonempty closed subsets of M. In general, the completion in the Hausdorff metric of the submodule lattice is isomorphic to the submodule lattice of the completion of the module in the J-adic metric. This extends submodule-theoretic results for complete modules to modules for which the submodule lattice is complete