Topologien von Moultonebenen. (Topologies of Moulton planes.) (Q1263082)

Let \((K,+,\cdot,<)\) be an ordered field, \(s\in K\), \(0<s\neq 1\). Define \(a\circ b:=a\cdot s\cdot b\), if \(a<0\), \(b<0\), else \(a\circ b:=a\cdot b\). Then \(K_ s=(K,+,\circ)\) is a Cartesian field, the corresponding projective plane is a (generalized) Moulton plane [cf. \textit{G. Pickert}, Projektive Ebenen (1975; Zbl 0307.50001), p. 93]. The author considers pairs \(T:=(\tau_ P,\tau_ L)\) of topologies on the point set P, resp. line set L of a Moulton plane \({\mathcal M}:=(P,L)\). If T is compatible \((=\) joining and interesecting are continuous mappings, i.e. (\({\mathcal M},T)\) is a topological projective plane), then \(\tau_ P\) and \(\tau_ L\) must contain the order topology (3), whence compact disconnected topological Moulton planes do not exist (5). As a main result (7) any topology on a Cartesian field \(K_ s\) rendering it a topological ternary field is extended to a compatible pair of topologies on \({\mathcal M}\). (3) is an important tool in the proof, the main steps of which are given; for a complete proof the reader is referred to the author, Topologisierung projektiver Ebenen durch Epimorphismen, Diss. Univ. München (1985; Zbl 0591.51021), p. 88-93. Let \(\lambda\) be a non-injective epimorphism defined on \({\mathcal M}\), and \(T_{\lambda}\) denote the pair of coarsest topologies on P, resp. L such that central projections are continuous and fibres of \(\lambda\) are open. \(T_{\lambda}\) is not discrete (15) (For projective planes in general, this has not yet been decided.), and \(T_{\lambda}\) induces on K the upper bound of the order topology and the place topology whose (sub)base is given by \(\{V_{ab}(x)|\) \(a,b\in K\}\), \(x\in K\) where \(V_{ab}(x):=\{\zeta \in K|\) \(\lambda (a(\zeta -x)b)=0\}\) (11) [cf. the author, Arch. Math. 51, No.3, 274-282 (1988; Zbl 0673.16023)]. Furthermore \(T_{\lambda}\) is compatible (11) [cf. the author, Geom. Dedicata 26, No.3, 259-272 (1988; Zbl 0659.51020)], 1.2 for a counterexample in a more general setting. The proof of (15) uses the fact that the place topology is a minimal ring topology, and that inverse complements of zero-neighbourhoods are s-bounded [cf. the author, loc. cit., (2.11), (2.12)].