Characterizing local connectedness in inverse limits
G. R. Gordh, Sibe Mardešić · Pacific Journal of Mathematics · 1975
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 σ-directed and surjective, then X is locally connected.(2) If X is well-ordered, surjective, and weight (X«) ^ λ for each a in A, then either weight (X) ^ λ, or X is locally connected.(3) If X is σ -directed and the factor spaces X α 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).