I came across this paper about Nearness Spaces. It seemed to be at the time (1970-80s) a promising approach to general topology via category theory. I have found no posts at all on stackexchange concerning Nearness Spaces. Has it hit a dead-end, been replaced by a better approach, simply forgotten, or called by a different name now?
As the author's references and abstract suggest, nearness spaces are a slight generalization of proximity spaces, an older concept.
There are a handful of Questions at Math.SE about proximity spaces.
Here are some "recent" research papers connected with nearness spaces:
Khare, Mona and Surabhi Tiwari: "Approach Merotopological Spaces and their Completion", Int. J. Math. and Math. Sciences (2010)
Leseberg, D.: "Supernearness Spaces as a Tool for Studying Unification and Extensions", Topology Atlas Invited Contributions 8 No. 3, 1–2 (2003)
Yang, Zhanbo: "A New Proof on Embedding the Category of Proximity Spaces into the Category of Nearness Spaces", Fundamenta Informaticae v.88 No. 1-2 (January 2008)
____ : "Pointwise convergence and Ascoli theorems for nearness spaces", Applied General Topology Vol 10, No 1 (2009)