Coordinate rings of topological Klingenberg planes I: The affine perspective

Abstract

Although the coordinate ternary field of a topological affine plane is topological, the converse does not hold. However, an affine plane is topological precisely when its coordinate biternary fields are topological. We extend this result to topological biternary rings and their topological affine Klingenberg planes. Then we examine the locally compact situation. Finally, following the ideas of Knarr and Weigand, we show that in certain circumstances, the continuity of the ternary operators is sufficient to ensure that the biternary ring is topological. This facilitates the construction of locally compact, locally connected affine Klingenberg planes.