This open research considers in details ordered semicategory actions (and ordered semigroup actions).
Ordered semicategory actions take a place in mathematics similar to "place" (generarity and amount of usage) of group theory. Therefore ordered semicategory actions are as important as group theory (like half of world economy). Contrary to sound sense, ordered semigroup actions (and ordered semicategory actions) were discovered (by the author) only in 2019, not in 19th century.
All kinds of spaces met in general topology, from locales/frames to metric spaces and everything in between, e.g. as topological spaces and uniform spaces (and, apparently, however yet unchecked, geometric spaces) are represented as elements of ordered semigroup with an action ("space-in-general"). This allows to study general topology in the universal, most general setting of space-in-general, namely as properties of ordered semicategory actions.
The theorems of general topology usually map to algebraic formulas with ordered semicategory actions, what much simplifies usage (and remembering) of formulas, because they are of an algebraic kind, not a mess of quantifiers as in traditional general topology.
So, called "funcoids" and also reloids, that is filters on Cartesian product of two sets, are of special interest and are researched in details. That's because funcoids generalize simultaneously: digraphs, topologies, pretopologies, preclosures, proximity spaces.
Also, they are researched "multidimensional" (including infinite dimensional) analogies of funcoids and reloids (next, should research multidimensional analogs of ordered semicategory actions).
Ordered semicategory actions are currently briefly considered at the end of the book, after their special cases (funcoids, reloids, etc.), because they were discovered when most of the book (about their special cases) has been already written. The book needs to be rewritten to consider ordered semicategory actions in more details, partly replacing the former text about their special cases.
The book also considers filters on ordered sets (and on lattices, powersets, etc.) in more details than any book before it.
The theory of funcoids (and ordered semicategory actions in general) is also used to describe discontinuous analysis, a generalized theory of limits where any function has a (generalized) limit at every point. This indespensibly simplifies physical calculations. This is probably also useful for general relativity and quantum gravity, as well as for economics, to name a few.
The primary author also created a big number of open problems a big part of which have been solved, but there remains quite a few.
The book about ordered semicategory actions is freely available.