Uniformly path connected homogeneous continua (Q2800407)

Two metric continua \(X\) and \(Y\) are continuously equivalent if there is an onto continuous mapping from \(X\) onto \(Y\) and there is an onto continuous mapping from \(Y\) onto \(X\). By the Hahn-Mazurkiewicz theorem and the Urysohn lemma, continua continuously equivalent to the interval \([0,1]\) are exactly the locally connected ones.NEWLINENEWLINEA metric continuum \(X\) is uniformly path connected if \(X\) is the continuous image of the cone over the Cantor set (called the Cantor fan). As usual, \(X\) is homogeneous, if for every pair of points \(p\) and \(q\) of \(X\), there exists a self homeomorphism of \(X\) taking \(p\) to \(q\).NEWLINENEWLINEThe main result of this paper says that:NEWLINENEWLINETHEOREM. Every arcwise connected homogeneous continuum is uniformly path connected.NEWLINENEWLINEAs a consequence of this theorem, the author obtains that every homogeneous arcwise connected continuum is either continuously equivalent to the unit interval or to the Cantor fan.NEWLINENEWLINEIn [Am. J. Math. 124, No. 4, 649--475 (2002; Zbl 1003.54022)] the same author constructed his important example of a non-locally connected, arcwise connected homogeneous continuum. So, indeed there exists an homogeneous continuum continuously equivalent to the Cantor fan.NEWLINENEWLINEThe continuum \(X\) has the arc approximation property if every subcontinuum of \(X\) is the limit, in the sense of the Hausdorff distance, of arcwise connected continua. The paper includes the following interesting question.NEWLINENEWLINEDoes every homogeneous arcwise connected continuum have the arc approximation property?