The present paper is concerned
with the lattice of regular topologies on a set, and establishes the following results: a
complete, complemented sublattice of the lattice of regular topologies on a set
is exhibited and shown to be anti-isomorphic to the lattice of equivalence
relations on the set; the lattice of regular topologies on a set is shown to
be nonmodular if the cardinality of the set is at least four; the problem of
complementation for regular topologies is reduced to considering T0 regular
topologies without isolated points; conditions are found which are equivalent
to a regular topology having a principal regular complement; then follow
some conditions under which the problem can be reduced to considering
connected spaces; the final section consists of constructions of complements
for certain classes of regular topologies, which classes may or may not be
exhaustive.
