A conjecture of Gleason on the foundations of geometry

Felix Klein emphasized the intrinsic connection between symmetry groups and geometries in his Erlangen Program. Perhaps motivated by Klein, Gleason [5] posed a very general conjecture on topologizing symmetry groups that he regarded as fundamental for a general study of geometries. Gleason in fact proved his conjecture in a very special case. The purpose of this paper is to show that Gleasonʼs general conjecture is false as originally stated and that it is true only under very strong hypotheses. Along the way new general results in descriptive set theory are proved about a class of functions that behave like but are distinct from functions of Baire class 1.