A new poset of topologies.

The family of all topologies on a given set , partially ordered by set inclusion, is a complete lattice in which the meet of a collection of topologies is their intersection and the join is the topology with their union as a subbase. In this talk, we first introduce a new partial order on and then we investigate some natural questions about the structure of this poset of topologies.

The research has been financed by the funding programme ``MEDICUS" of the University of Patras, Greece.