I think the following papers gives some ideas about current trend
[Lecture Notes in Computer Science] Rough Sets and Current Trends in Computing Volume 3066 || Spatial Object Modeling in Intuitionistic Fuzzy Topological Spaces
Tsumoto, Shusaku, Słowiński, Roman, Komorowski, Jan, Grzymała-Busse, Jerzy W.Year:2004
Generated fuzzy metric spaces and some topological properties
Çevik, Cüneyt, Eminoğlu, ŞehlaJournal:Journal of Intelligent & Fuzzy SystemsYear:2018
The applications of topology (General, Algebraic and Differential) appear in all branches of sciences. As Networking, Computer science, Graph theory, Dynamical systems, Physics, Chemistry, Biology(DNA codes), Economics, Cosmology, Combinatorics, and Economics, etc.
The literature is rich in unlimited publications about topology in all domains,
The current areas of researches in topology depend widely on geometric topology, algebraic topology and differential topology and the modern area now a days is the applied topology.
The current trend is towards topological manifolds (they are also called Riemannian manifolds), mainly due to the reason that such topological manifolds are locally Euclidean, and applicable to computer vision applications, such as classification, clustering, face recognition, object tracking and segmentation.