The study of relations between two topologies on the same set is a classical and quite old subject in General Topology with the partial order by inclusion being one of the most essential relations. Various relationships between topologies have been studied (cf. [A], [C], [D], [W]). In [M] we introduced the strongly finer relation “⊴” between two topologies on a given set X. This relation defines a new order on the family T(X) of all topologies on X, which is stronger than the usual subset relation. In this talk, we continue the investigation of the poset (T(X),⊴). In addition, we introduce some new relations between topologies on X.

References
[A] J. M. Aldaz, Uniformly finer topologies, Rend. Circ. Mat. Palermo (2) 45 (1996), no. 3, 453–458.
[C] Vitalij A. Chatyrko, On π-compatible topologies and their special cases, Topology Appl. 374 (2025), Paper No. 109243, 10 pp.
[D] B. P. Dvalishvili, Bitopological spaces: theory, relations with generalized algebraic structures, and applications, North-Holland Mathematics Studies, 199. Elsevier Science B.V., Amsterdam, 2005.
[M] A. C. Megaritis, A new poset of topologies, Mathematica Slovaca (2026), in press.
[W] J. D. Weston, On the comparison of topologies, J. London Math. Soc. 32 (1957), 342–354.