Topological geometric lattices (Q921024)

In two earlier papers of the second author geometric lattices G with a topology \({\mathcal T}\) were investigated, with compatibility conditions partly weaker than the continuity of the join and of the meet. It is natural to ask, why not simply require (G,\({\mathcal T})\) to be a topological lattice? The present paper answers this question: This requirement would be too strong, leaving only trivial topologies, and the Alexandrov topology and its dual.