On Zero-dimensional Continuous Images of Compact Ordered Spaces

Throughout this paper X denotes a compact and zero-dimensional Hausdorr space. We shall be concerned with Theorem 0.1 The following assertions are equivalent. (A) X is the continuous image of a compact ordered space. (B) X is the continuous image of a zero-dimensional compact ordered space. (C) X has a T 0-separating cross-free family of clopen sets. (D) X has a T 0-separating non-Archimedian family of clopen subsets. (E) X can be embedded as a strong T-set into a dendron. (F) X can be embedded into a dendron. (G) X has a cross-free closed subbase. Although probably never arranged in single le, the above equivalences are more or less known. The implication G) F is due to van Mill and Wattel (cf. 7, theorem 6.6]) and does not depend on X being zero-dimensional (but the general case it is much harder to prove). F) A follows from the fact that every dendron is a continuous image of a compact ordered space (a result due independently to J. L. Cornette 3] and A. E. Brouwer 2] see also 7, theorem 3.3]). The implication A) E is due to J. Nikiel 9]. S. Purisch recently observed E) D, which he combined with Nikiel's result to obtain A) D (cf. 12]), thus solving a problem posed by van Douwen in 4] and in Boolean algebraic form also by Koppelberg and Monk in 6]. The equivalence of F and H does not come out of the blue either. But it is not clear how to get it directly from L. E. Ward's result in 11]. In this note we ooer new proofs of most of the implications according to the following schema. E t =) F =) G 3 =) C 2 =) D 5 =) H 2 =) D 5 =) E k _ B t =) A 2 =) B 4 =) C; where trivial implications are marked by t =) and i =) means that the implication is proved in section i. The two unmarked implications will not be proved here, because we have no simpliications to ooer. In 7] the very rst thing van Mill and Wattel prove about dendrons is the fact that 1