  Lindenhovius has studied Grothendieck topologies on posets and has given a
complete classification in the case that the poset is Artinian. We extend his
approach to more general posets, by translating known results in locale and
domain theory to the study of Grothendieck topologies. In particular, explicit
descriptions are given for the family of Grothendieck topologies with enough
points and the family of Grothendieck topologies of finite type. As an
application, we compute the cardinalities of these families in various
examples.
