A new poset of topologies

The family <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">T</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> T(X) of all topologies on a given set X , 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 paper, we first introduce a new partial order on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">T</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> T(X) and then we investigate some natural questions about the structure of this poset of topologies.