Introduced several new axiomatic systems not less general than group theory and discontinuous analysis.
In this work I introduce and study in details the concepts of funcoids which generalize proximity spaces and reloids which generalize uniform spaces, and generalizations thereof. The concept of funcoid is generalized concept of proximity, the concept of reloid is cleared from superfluous details (generalized) concept of uniformity.
Also funcoids and reloids are generalizations of binary relations whose domains and ranges are filters (instead of sets). Also funcoids and reloids can be considered as a generalization of (oriented) graphs, this provides us with a common generalization of calculus and discrete mathematics.
I consider (generalized) limit of arbitrary (discontinuous) function, defined in terms of funcoids. Definition of generalized limit makes it obvious to define such things as derivative of an arbitrary function, integral of an arbitrary function, etc. It is given a definition of non-differentiable solution of a (partial) differential equation. It’s raised the question how do such solutions “look like'” starting a possible big future research program.
The generalized solution of one simple example differential equation is also considered.
The generalized derivatives and integrals are linear operators. For example $\int_a^b f(x)dx – \int_a^b f(x)dx = 0$ is defined and true for every function.
The concept of continuity is defined by an algebraic formula (instead of old messy epsilon-delta notation) for arbitrary morphisms (including funcoids and reloids) of a partially ordered category. In one formula continuity, proximity continuity, and uniform continuity are generalized.
Also I define connectedness for funcoids and reloids.
Then I consider generalizations of funcoids: pointfree funcoids and generalization of pointfree funcoids: staroids and multifuncoids. Also I define several kinds of products of funcoids and other morphisms.
I define space as an element of an ordered semigroup action, that is a semigroup action conforming to a partial order. Topological spaces, uniform spaces, proximity spaces, (directed) graphs, metric spaces, etc. all are spaces. It can be further generalized to ordered precategory actions (that I call interspaces). I build basic general topology (continuity, limit, openness, closedness, hausdorffness, compactness, etc.) in an arbitrary space. Now general topology is an algebraic theory.
Before going to topology, this book studies properties of co-brouwerian lattices and filters.