Vol. 58, No. 2, 1975

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Characterizing local connectedness in inverse limits

George Rudolph Gordh, Jr. and Sibe Mardesic

Vol. 58 (1975), No. 2, 411–417

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).

Mathematical Subject Classification 2000
Primary: 54B25
Secondary: 54F15
Received: 25 February 1974
Published: 1 June 1975
George Rudolph Gordh, Jr.
Sibe Mardesic