250 problems in elementary number theory by Waclaw Sierpinski

By Waclaw Sierpinski

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In contrast, here, we call a point set a neighborhood whenever we can find all points of a sufficiently small ball neighborhood in it. Given two sets 9l and '8 we define their set theoretic union 9l + 23 to be the set of those points which belong either to set 9l or to set 23 or to both sets. We define the set theoretic intersection of sets '3 and '8 to be the set of those points which belong to both 9l and to '8. If the intersection of two sets is empty then we say that the sets are disjoint. If % is a nonempty subset of a neighborhood space '332 we will define, once and for all, the neighborhood U(Q I Yt) of a point Q of 8 to be the intersection of any neighborhood U(Q I 2T) with %.

One may select an appropriately chosen representative from each class in such a way that these representatives comprise the interior points of a rectangle to which a corner point and two sides are added (Fig. 30), that is, all points whose coordinates satisfy the inequalities 0 Ix < a and 0 i y < b. When one now “identifies” equivalent points of the Euclidean plane the torus is generated, since the side x = a is equivalent to the side x = 0 and the side y = b is equivalent to the sidey = 0. Finally, we want to use the procedure of identification to construct surfaces by joining the sides of polygons.

14 1. ILLUSTRATIVE MATERIAL If we slice the one-sided tube which results along its length, it will split into two Mobius bands, one of which is illustrated in Fig. 23. When we attach k Mobius bands to the 2-sphere we obtain a closed surface, which can be given the fundamental polygon having boundary circle alala2a2. akak. We are not sure, yet, whether the closed surfaces which we have generated exhaust all of the closed surfaces, nor have we proven that no two of them can be mapped topologically one onto the other.

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