Arc- and Tree-preserving Transformations

1. Introduction. In an earlier paper(2) by one of us, referred to hereafter as A.P.T., arc-preserving transformations were defined and studied in connection with an irreducibility condition on the transformation. It was shown, for example, that if A and B are compact locally connected metric continua which are cyclic (that is, without cut points) any single valued continuous arc-preserving and irreducible transformation T(A) =B of A onto B is necessarily a homeomorphism. ("Arc-preserving" means that the image of every simple arc in A is either a simple arc or a single point of B; irreducibility of T means that no proper subcontinuum of A maps onto all of B.) It was shown, furthermore, that in case A is hereditarily locally connected the same conclusion holds without the assumption of irreducibility; and the prediction was made that this is true in the general case. Now as pointed out in A.P.T., if A is a compact continuum and T(A) =B is continuous, then, since the property of being a subcontinuum of A mapping onto all of B under T is inducible, there always exists a subcontinuum A\ of A such that T(Ai)=B and T is irreducible on Ai. However, since local connectedness of A would certainly not in general insure local connectedness of Ai, it is not possible always to reduce the set A so as to make the transformation irreducible without sacrificing essential properties of A. In the present paper we propose not only to completely justify the earlier prediction referred to above, but also to obtain theorems concerning a much more general type of transformation than "arc-preserving" which will give all the theorems of the first three sections of A.P.T. as immediate corollaries. R. G. Simond(3) has studied tree-preserving transformations on locally connected compact and metric continua (that is, transformations T(A)=B satisfying the condition that the image of every tree (or dendrite) in A is a tree in B). Miss Simond has proved with considerable difficulty that every arc-preserving transformation is tree-preserving. We show that this result is an immediate consequence of one of our theorems, as it is also of a theorem of A.P.T. In fact our first principal result, the proof of which is very much simpler than that given by Simond, shows that in order that T(A)=B be tree-preserving it is necessary and sufficient that the image of every simple