Continuous incidence relations of topological planes

Abstract

0. In topological planes the link between the geometrical and the topological structure is established by the requirement that the partial external operations "joining of points" and "intersecting of lines" be (op-) continuous (for the terminology, see Section 1). It seems that nowhere in the literature one has discussed the question, if and under which conditions the incidence relation (or other relations derived from it) underlying to the notion of a topological plane is (resp. are) "continuous" in some natural sense, or whether, in a more general setting (see Remark 6), the incidence relation should be required to be "continuous". In this paper, it turns out that the incidence relation of each topological plane and its inverse are lowersemicontinuous and they are continuous (in the sense introduced by the author in [3]) in certain classes of topological planes (Theorem 4, Propositions 5 through 8). As a byproduct in this paper, we characterize (in Theorem 1) the continuity of certain relations between a topological space and a product of topological spaces (which has a simple consequence in Proposition 9) and describe (in Theorem 3) the topology of the set of lines (of a topological plane) in terms of limits (via the notion of the power of a topology).