On ordered algebras
Alexa A. Albert · Bulletin of the American Mathematical Society · 1940
ORDERED ALGEBRAS 521 call it g' distinct from e. Let U be an open three-cell subset of R which includes e.Let N be a two-sphere made up of elements of G.The existence of arbitrarily small two-spheres of this kind is proved as above by choosing the element g sufficiently near to e, and we may assume that N is in U. We may also assume that LN is in U and that g'N is outside N: as must be the case if N is small enough.The arc L may now be used to define a deformation of N to g'N.Under this deformation all points swept out by N are in G. Furthermore every point inside N is swept out by the deformation.Hence every point of R inside N is in the group G.The group G is thus seen to contain open subsets and, because of homogeneity, G is open in R. It must therefore coincide with R. The assumption that a proper subgroup G was transitive on 5 has now led to a contradiction, and the proof is therefore complete.