Concerning a set of postulates for plane analysis situs

Abstract

My paper On the foundations of plane analysis situst contains three sets of postulates, 24, 22, and 3, expressed in terms of the undefined notions point and region. In the present paper I will show that every space S that satisfies :1 or 22 is a number plane, that is to say there exists, between S and a twodimensional euclidean space S', a one-to-one correspondence that preserves limits.t This signifies that if P is a point and M is a point-set in S, and P' and M' are the corresponding point and point-set in S', then P is a limit point of M in the sense defined on page 132 of the above mentioned paper if, and only if, P' is a limit point of M' in the sense that every circle in S' that encloses P' encloses also a point of M' distinct from P'. It follows that 24 and :2 are both categorical with respect to ? point and limit point as defined on page 132. Moreover between every space S, satisfying 21, and a two-dimensional euclidean space S' there exists a one-to-one correspondence preserving point and region if in S' the term region is interpreted to mean Jordan region. That is to say if the set of points M is a region in S then the set 1M1' of corresponding points in S' is the interior of a simple closed curve and conversely. Thus 21 is absolutelyll categorical. The system 22 is satisfied if in ordinary euclidean space of two dimensions the term region is interpreted as signifying Jordan region. It is however also satisfied if in such a space region is so interpreted