Continuous Collections of Decomposable Continua on a Spherical Surface

In this paper a study is made of continuous collections of decomposable continua on a spherical surface. Properties of the decomposition spaces of such collections filling up continua are established, and a characterization of the decomposition spaces of such collections filling up a spherical surface is obtained. The results of the present paper are related to certain results obtained by R. D. Anderson. He has shown [2] that there is a continuous collection of nondegenerate continua filling up the plane which is with respect to its elements homeomorphic to the plane, and that there is a continuous collection of continua filling up a plane onedimensional continuous curve which is with respect to its elements homeomorphic to the plane. On the other hand, he has obtained [l] a characterization of the plane decomposition spaces of continuous collections of nondegenerate continuous curves filling up a plane continuum. Such decomposition spaces are special types of hereditary continuous curves. The results of the present paper show that the decomposition spaces of continuous collections of decomposable continua filling up compact plane continua are special types of hereditary continuous curves, and consequently that even under this weaker hypothesis such collections are dimension reducing if they fill up twodimensional continua and are dimension preserving if they fill up one-dimensional continua. It might be noted that there do exist such collections filling up one-dimensional continua, but that Theorem 2 of this paper gives a strong restriction about the nature of such collections. Theorems about continuous collections of continua in a compact metric space can be stated equivalently in terms of monotone interior transformations (p. 130 of reference [lO]). It is understood throughout this paper that space is compact and metric. Definitions. If H is a point set and e is a positive number, V(e, H) denotes the set of all points P such that some point of H is at a distance from P of less than e. If H and K are point sets, d(H, K) will be used to denote the greatest lower bound of the set of all positive numbers e such that K intersects V(e, H). For two