Abstract
The family
T
(
X
)
$\mathcal{T}(X)$
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
T
(
X
)
$\mathcal{T}(X)$
and then we investigate some natural questions about the structure of this poset of topologies.