ARCWISE CONNECTIVITY AND CONTINUUM CHAINABILITY

ABSTRACT. A continuum is said to be continuum chainable provided that, for each pair x,y of points and each e> 0, there exists a chain of subcontinua from x to y such that each link has diameter less than e. While this generalization of property S is weaker than arcwise connectivity (even in the plane), it is shown that continuum chainability and arcwise connectivity are equivalent in each of the following classes of continua: hereditarily unicoherent continua, continua irreducible about a finite set, atriodic continua, and plane continua that do not separate the plane. 1. Definitions and preliminary remarks. A continuum is a compact connected metric space. A continuum M has propert) ' S [17, page 20] provided that lbr each e>0, M is the union of a finite collection of connected sets, each of diameter less than e. From compactness and the fact that property S is equivalent to local connectivity [17, page 23], it follows that a continuum with property S can be covered with arbitrarily small connected open sets, and hence, by [6, page 108], will have the property that for each pair x,y and each e>0, there exists a chain of connected sets from x to y with each link of diameter less than e. Replacing the words "connected sets " with the word "continua, " we obtain the notion of contimtum chai•bility. This property was introduced by Ira Rosenholtz. A finite collection (A 1..... A n} of sets is a chain from x to y provided x belongs only to A 1, y belongs only to A n, and for i,j G {1..... n}, A i C3Aj 4=4> if and only if li-jl•<l. If e>0 and x,y are distinct points of a continuum M, then by an e-continuum chain from x to y we mean a chain C from x to y such that each link (member of C) is a subcontinuum of M with diameter less than e. Thus M is continuum chainable if and only if tbr each pair x,y and each e>0, there exists an e-continuum chain from x to y