Mapping properties of weakly arcwise open dendroids

Abstract

A dendroid X is said to be weakly arcwise open if for each point p of X each arc component of X \ {p} either is open or has empty interior. We study various mapping properties of these dendroids. The leading problem is what classes of mappings between dendroids preserve the property of being weakly arcwise open.A dendroid means an arcwise connected and hereditarily unicoherent metric continuum. Investigation of shore points (special kind of noncut points) in dendroids, see [11,13], leads in a natural way to establish a class of dendroids called weakly arcwise open (WAO), being a generalization of dendroids having the property of Kelley, see [12]. The class of WAO dendroids was studied in [12], [13, Section 3], [14,3]. The aim of this paper is to continue this study, especially with respect to mapping properties of WAO dendroids. PreliminariesA continuum means a compact connected metric space. A continuum is said to be hereditarily unicoherent provided that the intersection of any two of its subcontinua is connected. A dendroid is an arcwise connected and hereditarily unicoherent continuum. Recall that each subcontinuum of a dendroid is a dendroid, and that for every two points a and b of a dendroid X there is a unique arc ab ⊂ X joining a and b.Given a subset S of a dendroid X, we denote by A(S) the family of all arc components of S in X. A point p of a dendroid X is called a ramification point (an end point) of X provided that card(A(X \ {p})) 3 (if card(A(X \ {p})) = 1, respectively). If a dendroid X has exactly one ramification point, then it is called a fan, and the ramification point is named the top of the fan.A dendroid X is said to be arcwise open (closed, respectively) at a point p ∈ X provided that each element of A(X \ {p}) is open (closed, respectively) with respect to the subspace X \ {p} of X. Since X \ {p} is an open subset * Corresponding person. E-mail address: sergiom@matem.unam.mx (S. Macias). Deceased. 0166-8641/$ -see front matter