Introduction

Abstract

In every textbook on the theory of relativity it is assumed, without any discussion, that space-time is a differentiable manifold, possibly with singularities. Between a point-set and a differentiable manifold there is an enormous gap, and we felt that physics itself could contribute to narrowing this gap. In experimental physics, one can make only a finite number of well-separated measurements, and therefore it is natural to start with a discrete set of points as a candidate for space-time. As one cannot put an upper bound on the number of measurements, this set cannot be finite; it must, at least, be countable. Next, as one cannot place a quantitative limit on experimental accuracy, one has to admit the “density” property that between any two points on a scale lies a third. Finally, one has to ask how one arrives at the continuum. In short, we felt that it should be possible to start from point-sets and find conditions (axioms) — motivated by physics — which would allow us to construct a topological manifold from the point-set. If this turned out to be true, one could become more ambitious and look for conditions which would imply the differentiability of the manifold.