Planar images of decomposable continua
Charles L. Hagopian · Pacific Journal of Mathematics · 1972
A nondegenerate metric space that is both compact and connected is called a continuum.In this paper it is proved that if M is a continuum with the property that for each indecomposable subcontinuum H of M there is a continuum K in M containing H such that K is connected im kleinen at some point of H and if / is a continuous function on M into the plane, then the boundary of each complementary domain of f(M) is hereditarily decomposable.Consequently, if M is a continuum in Euclidean w-space that does not contain an indecomposable continuum in its boundary, then no planar continuous image of M has an indecomposable continuum in the boundary of one of its complementary domains.For a given set Z, the closure and the boundary of Z are denoted by Cl Z and Bd Z respectively.The union of the elements of Z is denoted by St Z.