On Local Connectedness of Absolute Retracts

One of the fundamental observations of theory of retracts says that any absolute retract for the class of all compacta is a locally connected continuum. However, when more restricted classes of spaces are considered, as for example hereditarily unicoherent continua, tree-like continua, dendroids, or hereditarily indecomposable continua, this is not necessarily the case (see e.g., [6, Corollaries 4 and 5, p. 181 and 183]). In this paper we show that, nevertheless, for a large number of classes of continua their absolute retracts remain locally connected. The concept of an AR space originally had been studied by K. Borsuk, see [2]. More recently AR-spaces for some classes of continua had been studied e.g., in [6], [7] and [8]. The aim of this paper is to continue this investigation for various classes K of spaces. We focus our attention on conditions concerning classes K implying that each member of AR(K) is locally connected. By a space we mean a topological space, and a mapping means a continuous function. Given a space X and its subspace Y ⊂ X, a mapping r : X → Y is called a retraction if the restriction r|Y is the identity. Then Y is called a retract of X. Let K be a class of compacta, i.e., of compact metric spaces. Following [3, p. 80], we say that a space Y ∈ K is an absolute retract for the class K (abbreviated AR(K)) if for any space Z ∈ K such that Y is a subspace of Z, Y is a retract of Z. The reader is referred to [2] and [3] for needed information on these concepts. By a continuum we mean a connected compactum. A curve means a onedimensional continuum. For a given continuum X, an arc component of X means the union of all arcs A such that p ∈ A ⊂ X for some point p of X. A locally connected continuum containing no simple closed curve is called a dendrite. A continuum is said to be decomposable provided that it can be represented as the union of two its proper subcontinua. Otherwise it is said to be indecomposable. A continuum is said to be hereditarily decomposable (hereditarily indecomposable) provided that each of its subcontinua is decomposable (indecomposable, respectively).