Vol. 43, No. 1, 1972

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Complementation in the lattice of regular topologies

M. Jeanette Huebener

Vol. 43 (1972), No. 1, 139–149

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.

Mathematical Subject Classification 2000
Primary: 54A10
Received: 15 July 1971
Revised: 6 September 1971
Published: 1 October 1972
M. Jeanette Huebener