A topological description of space is given, based on the relation of connection among regions and the property of being limited. A minimal set of 10 constraints is shown to permit definitions of points and of open and closed sets of points and to be characteristic of locally compact T2 spaces. The effect of adding further constraints is investigated, especially those that characterise continua. Finally, the properties of mappings in region-based topology are studied. Not all such mappings correspond to point (...) functions and those that do correspond to continuous functions. (shrink)

Whitehead, in his famous book Process and Reality, proposed a definition of point assuming the concepts of "region" and "connection relation" as primitive. Several years after and independently Grzegorczyk, in a brief but very interesting paper, proposed another definition of point in a system in which the inclusion relation and the relation of being separated were assumed as primitive. In this paper we compare their definitions and we show that, under rather natural assumptions, they coincide.

Whitehead, in his famous book "Process and Reality", proposed a definition of point assuming the concepts of “region” and “connection relation” as primitive. Several years after and independently Grzegorczyk, in a brief but very interesting paper proposed another definition of point in a system in which the inclusion relation and the relation of being separated were assumed as primitive. In this paper we compare their definitions and we show that, under rather natural assumptions, they coincide.

Although Aristotle (Metaphysics, Book IV, Chapter 2) was perhaps the first person to consider the part-whole relationship to be a proper subject matter for philosophic inquiry, the Polish logician Stanislow Lesniewski [15] is generally given credit for the first formal treatment of the subject matter in his Mereology.1 Woodger [30] and Tarski [24] made use of a specific adaptation of Lesniewski's work as a basis for a formal theory of physical things and their parts. The term 'calculus of individuals' was (...) introduced by Leonard and Goodman [14] in their presentation of a system very similar to Tarski's adaptation of Lesniewski's Mereology. Contemporaneously with Lesniewski's development of his Mereology, Whitehead [27] and [28] was developing a theory of extensive abstraction based on the two-place predicate, 'x extends over y\ which is the converse of 'x is a part of y\ This system, according to Russell [22], was to have been the fourth volume of their Pήncipia Mathematica, the never-published volume on geometry. Both Lesniewski [15] and Tarski [25] have recognized the similarities between Whitehead's early work and Lesniewski's Mereology. Between the publication of Whitehead's early work and the publication of Process and Reality [29], Theodore de Laguna [7] published a suggestive alternative basis for Whitehead's theory. This led Whitehead, in Process and Reality, to publish a revised form of his theory based on the two-place predicate, 'x is extensionally connected with y\ It is the purpose of this paper to present a calculus of individuals based on this new Whiteheadian primitive predicate. (shrink)