The lattice of closed congruences on a topological lattice
Dennis J. Clinkenbeard · Transactions of the American Mathematical Society · 1981
Our primary objectives are: (1) if L L is a lattice endowed with a topology making both the meet and join continuous then (i) the natural map which associates a congruence with the smallest topologically closed congruence containing it preserves finite meets and arbitrary joins; (ii) the lattice of such closed congruences is a complete Brouwerian lattice; (2) if L L is a topological (semi) lattice with the unit interval as a (semi) lattice homomorphic image then the lattice of closed (semi) lattice congruences has no compatible Hausdorff topology.