A Cyclic Element Characterization of Monotone Normality

Abstract

A subcontinuum g of a locally connected continuum X is a cyclic element of X provided that g is maximal with respect to the property that no point separates it. In an earlier paper, Cornette showed that a locally connected continuum is the continuous image of an arc if and only if each cyclic element of X is the continuous image of an arc. In this paper we prove the analogous theorem for monotonically normal continua by showing that a locally connected continuum X is monotonically normal if and only if each cyclic element of X is monotonically normal.Definition. A continuum is a compact connected Hausdorff space. A continuum is called an arc provided that it is a nondegenerate ordered continuum.Notation. If S ⊂ X, Int X (S) will denote the interior of S with respect to X or simply Int (S) if the superspace is clear. Similarly, ∂ X (S) or ∂(S) will denote the boundary of S with respect to X. Definition.A cyclic element C of a locally connected continuum X is a subcontinuum of X that is maximal with respect to the property that no point separatescontains at most one point, and where if C is an open cover of X then all but a finite number of the G i lie in some element of C. For any two distinct points a and b of X, the intersection of all A-sets in X containing a and b is called the cyclic chain from a to b and is denoted by C(a, b).