On the generation of a simple surface by means of a set of equicontinuous curves

The purpose of this article is to prove: Theorem: Suppose that, in a given three dimensional space S, ABCD is a rectangle and G is a self-compact set of simple continuous arcs such that: 1. through each point of ABCD there is just one arc of G, 2. BC and AD are arcs of G, 3. no two arcs of G have a point in common, 4. each arc of G has one endpoint on the interval AB and one endpoint on the interval CD but contains no other point in common with either of these intervals, 5. the set of arcs G is equicontinous. Then the point - set R composed of all the arcs of the set G is in one to one continuous correspondence with the plane point-set formed by a rectangle together with its interior.