Connected open sets in products of indecomposable continua
by
David P. Bellamy
University of Delaware
Coauthors: Janusz M. Lysko, Widener University

This is a report on ongoing work. Several results will be announced, along with a proof of at least one of them. The following theorem was the beginning of this study:

Theorem. In the product of a pseudo-arc with itself, the diagonal has arbitrarily small connected open neighborhoods.

This result also holds for any Knaster-type indecomposable continuum in place of the pseudo arc. The proof for the Knaster continuum case is completely different from the pseudo arc case. It is not clear whether this is true for every chainable indecomposable continuum, but there exist indecomposable continua, notably solenoids, which do not have this property.

We have a number of stronger theorems, including theorems about products of more than two factors, and products of nonhomeomorphic factors. There are also many unanswered questions.

Date received: February 28, 2005