Analytic transformations of everywhere dense point sets

Abstract

There is a well known theorem in the theory of point sets, due to Cantor,t to the effect that All enumerable, everywhere dense linear point sets without first and last points have the same order type as the rational numbers. That is, any set of this type can be mapped on the rational points of a line by a one to one correspondence which preserves order, and consequently any two sets of this type can be mapped on one another by such a correspondence. A correspondence of two everywhere dense point sets clearly determines at most one continuous function which maps the segments on which the given sets are everywhere dense on one another, and also generates the correspondence. The requirement that the correspondence preserve order is equivalent to the requirement that a continuous mapping function exist, so that we may state the above theorem in the following form: For any two enumerable linear point sets, each everywhere dense on an open interval, a continuous function can be found which maps the two intervals on one another, and effects a onte to one correspondence between the point sets. Since the function of this theorem is by no means uniquely determined, the question naturally arises as to whether we can place further restrictions on it without destroying the validity of the theorem. It turns out that we may always require the function which effects the mapping to be analytic and it is the demonstrationi of this fact and some related questions which occupy our attention in this paper.