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.

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.

Before going to topology, this book studies properties of co-brouwerian
lattices and filters.

## Social share