On non-metric indecomposable and hereditarily indecomposable subcontinua of products of long Hausdorff arcs.
by
Michel Smith
Auburn University

Consider the long line L with endpoints 0 and w₁. We show that every indecomposable subcontinuum of L² is metric but that L³ contains a non-metric indecomposable continuum. This is not surprising but the situation for hereditarily indecomposable continua is much more interesting. Let A be a non-metric Hausdorff arc such that the set Q of w₁-type of points is no-where dense and such that for each point p in A - Q there is a metric open set containing p (as is the case with the long line), then every hereditarily indecomposable subcontinuum of Aⁿ is metric for n finite or countable. We also observe that if A is a non-metric Hausdorff arc then A³ contains a non-metric indecomposable continuum.

Date received: February 1, 2006