On topological Möbius planes in reflection-geometric representation (Q1122780)

For the special case of a miquelian Möbius plane (inversive plane) \({\mathcal M}\) satisfying the theorem of tangent pencil the author gave a group theoretical axiom system [Stud. Sci. Math. Hung. 21, 363--372 (1986; Zbl 0585.51006) and Intuitive geometry, Pap. Int. Conf., Siofok/Hung. 1985, Colloq. Math. Soc. János Bolyai 48, 617--629 (1987; Zbl 0638.51009)] in terms of the set \({\mathcal S}\) of all reflections at circles of \({\mathcal M}\) (elements of \({\mathcal S}\) represent circles of \({\mathcal M})\) and a covering \({\mathcal P}\) of invariant subsets of \({\mathcal S}\) (elements of \({\mathcal P}\) represent points of \({\mathcal M})\). Similar as earlier for a reflection-geometric representation of affine Moufang planes [the author, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 29, 41--62 (1986; Zbl 0623.51008)] in the present note topological conditions to the author's former reflection-geometric axiom system are added such that precisely topological Möbius planes in the sense of \textit{R.-D. Wölk} [Math. Z. 93, 311--333 (1966; Zbl 0143.44001)] are obtained (however Wölk deals with general not necessarily miquelian Möbius planes). As in Wölk's axiom system two of the four topological axioms require that the set of all pairs of circles intersecting in two points is open in \({\mathcal S}^2\) and that the operations of joining three points by a circle and intersecting two circles are continuous. Furthermore the topology of \({\mathcal P}\) which is obtained from the topology of \({\mathcal S}\) in a complicated process has to be separated (it is shown subsequently that the topology of \({\mathcal S}\) is \(T_1)\), the elements of \({\mathcal S}\) have to operate continuously on \({\mathcal P}\) and a coherence axiom is required. Using orthogonality Wölk's topological axioms are derived. Conversely, a topological miquelian Möbius plane satisfies the author's axioms in the reflection-geometric description [see the author's doctoral thesis ``Topologische affine Ebenen und topologische Möbiusebenen in spiegelungsgeometrischer Darstellung'', Potsdam (1985)].