Weaker Versions of Zeeman's Conjectures on Topologies for Minkowski Space

Abstract

In a paper [Topology 6, 161 (1967)] Zeeman conjectured that (a) the finest topology on Minkowski space that induces the one‐dimensional Euclidean topology on every timelike line and (b) the finest topology on Minkowski space that induces the three‐dimensional Euclidean topology on every spacelike hyperplane have the same group of homeomorphisms G which is generated by the inhomogeneous Lorentz group and the dilatations. This paper deals with two topologies on Minkowski space which are weaker than those in (a) and (b), respectively, and have the property that they induce the one‐dimensional and the three‐dimensional Euclidean topology on timelike lines and spacelike hyperplanes, respectively. It is then shown that both topologies have G as their homeomorphism group. Thus, what we have shown amounts to proving the weaker versions of Zeeman's conjectures.