ABSTRACT
Throughout this paper, by a continuum we mean a compact, connected subset of a metric space and by a mapping we mean a continuous function. A continuum is decomposable if it is the union of two of its proper subcontinua and it is indecomposable, otherwise. If Xi, X-z, X%, • • • is a sequence of topological spaces and /i, /2, /a, • • • is a sequence of mappings such that, for each i, fi : Xi+i — > Xi, by the inverse limit of the inverse sequence {Xi, f i } , denoted lim{Xj, f i } , is meant the subset of f| Xi to which the point x belongs if and only if fi(xi+i) = Xi for i = 1, 2, 3, • • • . It is well known that when the spaces Xi are continua, the inverse limit is a continuum. If, for each positive integer i, Xi is a continuum, the statement that the inverse sequence {Xi, fi} is an indecomposable inverse sequence means that, for each positive integer i, if Xi+i is the union of two subcontinua Ai+i and Bi+i then fi[Ai+i] = Xi or fi[Bi+i] = Xt. In [4, Theorem 2.7, page 21], it is shown that if { X i , f i } is an indecomposable inverse sequence, then lim{Xj,/j} is an indecomposable continuum. If for each positive integer i, Xi = X and fi = f, we shall denote the inverse limit by lim{j!f, /}. We denote the natural projection of the inverse limit to the nth factor space by ?rn and if if is a closed subset of the inverse limit space, we often denote nn[H] by Hn. If P is a point of a continuum M with the property that if each of H and K is a subcontinuum of M containing P then H is a subset of K or K is a subset of H, we call P an end point of M.
