Bend intersection property and dendroids of type N (Q1182542)

It is well known that if a curve (i.e. one-dimensional continuum) is contractible, it must be a dendroid (see [\textit{J. J. Charatonik}, Gen. Topol. Relat. mod. Anal. Algebra, Proc. 4th Prague Topol. Symp. 1976, Part B, 72-76 (1977; Zbl 0373.54029), p. 73]). In the contrary to fans [\textit{L. G. Oversteegen}, Bull. Acad. Pol. Sci., Ser. Sci. Math. 27, 391- 395 (1979; Zbl 0439.54033)] no internal characterization of contractible dendroids is known till now. Several either necessary or sufficient conditions under which a dendroid is contractible are dispersed in the literature. One of them is of being of type \(N\), a condition formulated in [\textit{L. G. Oversteegen}, ibid., Ser. Sci. Math. Astron. Phys. 26, 837-840 (1978; Zbl 0404.54031)] proved to be sufficient for non- contractibility of an arbitrary metric space, and exploited later to characterize contractible fans. The aim of this paper is to characterize the property of being of type \(N\) for dendroids by the so called bend intersection property. This property, introduced and studied by \textit{T. Maćkowiak} [ibid., 547-551 (1978; Zbl 0412.54018)] for dendroids which admit a continuous selection on the hyperspace of their subcontinua, has appeared to be a good tool also in investigations of internal structure of contractible fans [the author, Bull. Pol. Akad. Sci., Math., 36, No. 7/8, 413-417 (1988; see the following review)]. It is proved that a dendroid \(X\) is not of type \(N\) if and only if for each arc of \(X\), the intersection of all its bend sets is nonempty.