A class of topological affine translation planes having no topological projective extension
For any quadrangle \((o,u,v,e)\) in a projective plane, one obtains an algebraic structure of the affine line \(ov\backslash\{v\}\), called ternary field. The ternary operation is defined by means of joining points and intersecting lines. This ternary field encodes the geometry; in fact, every ternary field determines a projective plane up to isomorphism. If the projective plane is topological (i.e., endowed with topologies that render the operations ``join'' and ``intersect'' continuous), then the ternary operation (and every geometrically relevant ``inverse'') is obviously continuous as well. However, the construction of a projective plane from a given ternary field starts with an affine point set and the set of all affine lines of finite slope, while the remaining points and lines are added afterwards. This implies that a topological ternary field need not give rise to a topological projective plane, not even to a topological affine plane [cf. the author, Topologische Divisionsalgebren ohne zugehörige topologische affine Ebene, Abh. Math. Semin. Hamb. 62, 167-177 (1992)]. Under additional topological or algebraical (resp. geometrical) assumptions, it has been proved that the projective plane is topological, see the references in the paper under review. In the present paper, a construction of D. Betten is generalized to obtain a class of topological quasifields \(Q\) (of dimension 2 over some ordered commutative field \({\mathbf K}\)) which yield topological (translation) planes (Theorem 2.2). The projective extension of such an affine plane is, in general, not topological; in fact, the author proves the following (Theorem 2.5). If \({\mathbf K}\) has an infinite transcendence basis over its prime field, then there is a valuation topology \(\mu\) of \({\mathbf K}\) such that the supremum of \(\mu\) and the order topology gives a field topology on \({\mathbf K}\) such that, with respect to the corresponding product topologies, the quasifield \(Q\) and the affine plane over \(Q\) are topological, but the projective extension is not. If \({\mathbf K}\) is a real closed field, then every real valuation topology \(\nu\) that is different from the order topology may be taken for \(\mu\) (Theorem 2.6). In the formulation of Theorem 1.4, the term ``topological direct sum'' should be replaced by ``topological direct product''.
scientific article; zbMATH DE number 428630