$\lambda$ connected plane continua

Abstract:

A continuum M is said to be ${\mathbf {\lambda }}$ connected if any two distinct points of M can be joined by a hereditarily decomposable continuum in M. Recently this generalization of arcwise connectivity has been related to fixed point ptoblems in the plane. In particular, it is known that every ${\mathbf {\lambda }}$ connected nonseparating plane continuum has the fixed point property. The importance of arcwise connectivity is, to a considerable extent, due to the fact that it is a continuous invariant. To show that ${\mathbf {\lambda }}$ connectivity has a similar feature is the primary purpose of this paper. Here it is proved that if M is a ${\mathbf {\lambda }}$ connected continuum and f is a continuous function of M into the plane, then $f(M)$ is ${\mathbf {\lambda }}$ connected. It is also proved that every semiaposyndetic plane continuum is ${\mathbf {\lambda }}$ connected.