I’ve published a new edition of my book Algebraic General Topology. The new edition features “unfixed morphisms” a way to turn a category into a semigroup. (Certain additional structure on the category is needed.)
The book features a wide generalization of general topology done in an algebraic way. Now we can operate on general topological objects with algebraic operations.
The book features the concept of funcoid – something better than topological space. Topological spaces mostly become history, the new thing of general topology is funcoids. It also considers filters on the Cartesian product of sets, pointfree topology, “multidimensional” general topology, where traditional topological spaces are always two-dimensional in a sense, as they consider binary relations (a relation between one point and one set).
Completely overthrow your understanding of general topology with this book. The series of the book also presents even more abstract semigroup algebra of general topology and a generalization of limit for an arbitrary function.