A Note on Inverse Limits of Continuous Images of Arcs

The main purpose of this paper is to prove some theorems concerning inverse systems and limits of continuous images of arcs. In particular, we shall prove that if X = {Xa, pab, A} is an inverse system of continuous images of arcs with monotone bonding mappings such that cf(card(A)) = ω1, then X = limX is a continuous image of an arc if and only if each proper subsystem {Xa, pab, B} of X with cf(card(B)) = ω1 has the limit which is a continuous image of an arc (Theorem 18). 1. Inverse limits of hereditarily locally connected continua An arc (or ordered continuum) is a Hausdorff continuum with exactly two non-separating points. Each separable arc is homeomorphic to the closed interval I = [0, 1]. A space X is said to be an IOK (IOC) if there exists an ordered compact (connected) space K and a continuous surjection f : K → X. Frequently, we will say that a space X is a continuous image of an arc if X is an IOC. The cardinality of a set A will be denoted by card(A). We assume that card(A) is the initial ordinal number. The cofinality of a cardinal number m will be denoted by cf(m).