Topology and Elementary Geometry. Ii. Symmetries1

Modern axiomatics of elementary geometry has steadily followed Hubert's program of elimination of continuity arguments. A brilliant example of the success of this method is to be found in the book [l ] by F. Bachmann which gives a unified treatment for geometries over arbitrary fields of characteristic y¿2. On the other hand the author has shown [2 ] that the plane axioms of incidence and order in Hubert's system have very strong topological implications. Therefore it seems natural to invert Hilbert's procedure and to try and characterize euclidean (and hyperbolic) geometry by the topology of its plane and some continuity properties of isometric mappings. This is done in this note. We give first a topological formulation of the order axioms and deduce the Hubert system from it. The topological approach allows some reduction of the axiom system. In the topological space generated we introduce symmetries. Since we dispose of a strong topological structure we can do with only part of Bachmann 's axioms. Finally we derive the congruence axioms as theorems. The main results are that completeness of the uniform structure and the existence of an unique involutive automorphism for each line imply the three-symmetries-theorem and the existence of angle bisector and perpendicular bisector, i.e., of a transitive group of motions. The topology of the plane x depends in large measure on the cardinality K„ of it. For a = 1 the plane is the cartesian product of two real number lines, it is two-dimensional. For ct>l it is totally disconnected and hence of dimension zero. Therefore it is interesting to note that we may put a cardinality axiom last in our list and never use it for the proof of congruence properties. This shows that the important topological properties in plane geometry are completeness and connectedness but not arcwise connectedness and dimension. We shall freely use the language of elementary geometry where no confusion can result.