On the Foundations of Plane Analysis Situs

Abstract

The present paper contains tbree systems of axioms, Z1, 12, and . Each of these systems is a sufficient basis for a considerable body of theorems in the domain of plane analysis situs or what may be roughly termed the nonmetrical part of plane point-set theory, including the theory of plane curves. The axioms of each system are stated in terms of a class of elements called points and a class of point-sets called regions. On the basis of Z1 the existence of simple continuous curves is proved as a theorem and it is shown that every region is the interior of a simple closed curve. 12 is equivalentt to Z1 as far as statements in terms of point and limit point are concerned. But 12 is satisfied if in an ordinary euclidean space of two dimensions the term region is interpreted so as to apply to every bounded, connected set of points R of connected exterior such that every point of R lies in the interior of some triangle that is contained in R. Both Z1 and 2 contain an axiom (Axiom 1) which postulates the existence of a countable sequence of regions containing a set of subsequences that close down in a specified way on the points of space. Among other things this axiom implies that the set of all points is separable.: The set 3 iS obtained from 2 by replacing Axioms 1 and 2 by two other axioms, Axioms 1' and 2'. Here Axiom 1' postulates the existence for each point P of a countable sequence of regions that closes down on P, while Axiom 2' postulates that every two points of a region are the extremities of at least one arc lying in that region. 12 implies 3 but not conversely. Though Theorems 1-52 of the present paper are all consequences of 3 nevertheless there exists a space that satisfies 3 but is neither metrical, descriptive? nor separable. It is interesting that no space that satisfies 3