  We discuss some notions of compactness and convergence relative to a
specified family F of subsets of some topological space X. The two most
interesting particular cases of our construction appear to be the following
ones.
  (1) The case in which F is the family of all singletons of X, in which case
we get back the more usual notions.
  (2) The case in which F is the family of all nonempty open subsets of X, in
which case we get notions related to pseudocompactness.
  A large part of the results in this note are known in particular case (1);
the results are, in general, new in case (2). As an example, we characterize
those spaces which are D-pseudocompact, for some ultrafilter D uniform over
$\lambda$.
