Let X denote the limit of an
inverse system X = {Xα;pαα′;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 pαα′ are suriections, then X is locally connected if and only if X is. The
following corollaries are obtained. (1) If X is σ-directed and surjective, then X is
locally connected. (2) If X- is well-ordered, surjective, and weight (Xα) ≦ λ for each
α in A, then either weight (X) ≦ λ, or X is locally connected. (3) If X is
σ-directed and the factor spaces Xα are trees (generalized arcs), then X is a
tree (generalized arc). (4) If X is well-ordered and the factor spaces Xα are
dendrites (arcs), then either X is metrizable, or X is a tree (generalized
arc).
