A Characterization of Smooth Cantor Bouquets

Abstract

We prove that all smooth fans having a dense set of endpoints are topologically equivalent.Let I bea smooth fan whose set of endpoints is dense in X.Such fans have been constructed, e.g., by J. H. Roberts [6], who proved that the space of rational sequences of the Hubert cube can be embedded in the Cantor fan, and by A. Lelek [3], who showed the existence of a fan whose (one-dimensional) set of endpoints can be connectified by adding the vertex.Lately, spaces similar to X \{v}, where v is the vertex of X, were discovered to be Julia sets of some nice analytic functions (see R. L. Devaney and M. Krych [2]; see also J. C. Mayer [4]).We are going to prove that all such examples are homeomorphic.Theorem.All smooth fans having dense set of endpoints are topologically equivalent.Let us recall that a continuum X is said to be hereditarily unicoherent if K n L is connected for every pair K , L of subcontinua of X .A continuum X is called a dendroid if it is arcwise connected and hereditarily unicoherent.By a fan we will mean a dendroid having exactly one ramification point; we will call this point the vertex of X .A fan X is said to be smooth if the sequence of arcs [w,xn] converges to the arc [v,x] for every sequence xn converging to x , where x, xn e X and v is the vertex of X.If X is a fan, then E(X) will denote the set of endpoints ofX.Ifx.yGÍHxíH, then by \x -y\ we will denote the Euclidean distance between points x and y and by [x, v] we will mean the linear segment with endpoints x and y . Mappings between inverse systemsThe following lemma is similar to [5, Theorem 2 ].The proof is a standard inductive argument and is left to the reader.