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Local cut-points in continuous images of compact ordered spaces

We prove that if a continuum X is continuous image of a compact ordered space and if X is not locally separable at a point x, then x lies in the closure of the set of all local cut-points of X. A continuum is a compact connected Hausdorff space. An ordered space is a totally ordered set endowed with the topology generated by open intervals. It is known from L. B. Treybig (4) that if a nonmetrizable continuum X is a continuous image of a compact ordered space, then some subset of less than three points separates X. In this paper we prove the following theorem by modifying Treybig's method, and give a corollary. Theorem. If a continuum X is a continuous image of a compact ordered space and if X is not locally separable at x0, then for any neighborhood U of x0 there exists a nonempty open set W such that W c U and the boundary Bd(W) consists of at most two points. A space X is said to be locally separable at x if there is a separable neighborhood of x. A point x of I is a local cut-point of X if there is a connected open neighborhood {/of x such that U — {x} is not connected. We will denote the set of all local cut-points of X by L{X). If IF is a nonempty open subset of a continuum X such that its closure W is not X, then its boundary Bd(fF) separates X. In addition, if Bd(lF) is finite, Bd(iF) contains at least one local cut-point. Hence, the theorem implies: Corollary. Let X be a continuum which is a continuous image of a compact ordered space. If X is not locally separable at x0, then x0 is in the closure of L(X). In the preceding corollary, even if X is locally connected, we can assert neither that x0 is always an accumulation point of L(X), nor that x0 is a local cut-point or an end-point of X. Counterexamples for these will be given in §3.