Characterizing local connectedness in inverse limits

Let X denote the limit of an inverse system X -{X a p aa >; A} of locally connected Hausdorff continua. The main purpose of this paper is to define a notion of local connectedness for inverse systems, and to prove that if X is locally connected, then so is the limit X. If the bonding maps p aa > are surjections, then X is locally connected if and only if X is. The following corollaries are obtained. (1) If X is -directed and surjective, then X is locally connected. (2) If X is well-ordered, surjective, and weight (X) ^ for each a in A, then either weight (X) ^ , or X is locally connected. (3) If X is -directed and the factor spaces X are trees (generalized arcs), then X is a tree (generalized arc). (4) If X is well-ordered and the factor spaces X are dendrites (arcs), then either X is metrizable, or X is a tree (generalized arc).