Comparison of Topologies (Minimal and Maximal Topologies)

Inherent to the study of topology is the notion of comparison of two different topologies on the same underlying set. This comparison is carried out by the natural ordering of set inclusion. Thus, the family of all topologies definable for an infinite set X is a complete atomic and complemented lattice (under set inclusion) which is denoted as LT (X). If and S are two topologies on X with S ⊆ T, then S is said to be weaker or coarser than , and is said to be stronger or finer than S. Given a topological invariant P, a member of LT (X) is said to be minimal or maximal P if and only if (iff) possesses the property P. However, in the former case, no other weaker and in the later case no other stronger member of LT (X) possesses the property P. In the same way, P is said to be expansive or contractive iff for each P-member of LT (X), every stronger or in the later case every weaker member of LT (X) is also P.