A Theorem on Continuous Decompositions of the Plane into Nonseparating Continua

E. Dyer [2] proved that there is no continuous decomposition of a compact irreducible continuum into decomposable continua which is an arc with respect to its elements. The author extends Dyer's result to the plane. Consider a continuous decomposition of the plane into nonseparating compact continua. R. L. Moore [6] has shown that the decomposition space is homeomorphic to the plane. Using Moore's result it is shown that the union of the elements of each arc in the decomposition space is an irreducible continuum. It follows then, from Dyer's result, that there is no continuous decomposition of the plane into nonseparating compact decomposable continua. In 1936 J. H. Roberts [7] proved that there is no upper semicontinuous decomposition of the plane into arcs. In the same paper Roberts gives an example of an upper semicontinuous decomposition of the plane each element of which is an arc or an "H". In 1968 Stephen L. Jones extended Roberts' theorem to E" [3]. R. D. Anderson announced [1] that there exists a continuous decomposition of the plane into pseudo-arcs. In 1949 E. E. Moise [5] proved that there is no continuous decomposition of a compact irreducible continuum into arcs which is an arc with respect to its elements. E. Dyer extended this result and showed that there is no continuous decomposition of a compact irreducible continuum into decomposable continua which is an arc with respect to its elements [2]. It follows from Dyer's result and the theorem proven in this paper that there is no continuous decomposition of the plane into nonseparating compact decomposable continua. If G is a collection of point sets then G* means the union of the elements of G. If G is an upper semicontinuous collection then G*/G means the decomposition space. The Euclidean plane is denoted by E2. The author wishes to thank the referee for indicating a much shorter proof of the main theorem which follows. The following theorems are assumed; for proofs the reader should refer to [6]. Theorem A. If G is an upper semicontinuous collection of compact continua Received by the editors April 14, 1975 and, in revised form, July 21, 1975. AMS (MOS) subject classifications (1970). Primary 54B15.