Disk-like products of $\lambda$ connected continua. II

Abstract:

R. H. Bing [3] proved that every atriodic, hereditarily decomposable, hereditarily unicoherent continuum is arc-like. Using this theorem, the author [5] showed that $\lambda$ connected continua $X$ and $Y$ are arc-like when the topological product $X \times Y$ is disk-like. In this paper we consider products that have a more general mapping property. Suppose that $X$ and $Y$ are $\lambda$ connected continua and that for each $\varepsilon > 0$, there exists an $\varepsilon$-map of $X \times Y$ into the plane. Then $X$ is either arc-like or circle-like. Furthermore, if $X$ is circle-like, then $Y$ is arc-like. Hence $X \times Y$ is either disk-like or annulus-like.