A Relation between Non-alternating and Interior Transformations

Recently in proving certain existence theorems for non-alternating and for interior transformations of a continuum onto a simple arc it was observed that when a transformation of one of these types was set up, usually it satisfied in large measure conditions which brought it also under transformations of the other type. This suggests the existence of common ground shared by these sorts of transformations, and it is the object of this paper to exhibit the nature of such. Our principal conclusion is to the effect that, under certain auxiliary conditions, any interior transformation ƒ(M) —D of a compact continuum M onto a dendrite D can be factored into the form f=f2fi where fi(M) = M' merely shrinks sets of type f~(p) (p an end point of D) into single points and is topological elsewhere and fî{M') =D is nonalternating and interior. However, we first prove two theorems on the relation between the non-alternating property of a transformation of a continuum into a dendrite and the connectedness of the inverse sets for the end points of the dendrite.