On generalized homogeneity of curves of a constant order (Q1335176)

A point \(x\) of a space \(X\) is said to have Menger-Urysohn order \(\kappa\) if \(\kappa\) is the smallest cardinal such that \(X\) has a local base at \(x\) of sets whose boundaries have cardinality \(\kappa\). If \(X\) has a local base at \(x\) of sets with finite boundary, with the supremum of the cardinalities of the boundaries being infinite, then the order of \(X\) at \(x\) is said to be \(\omega\). A space \(X\) is said to be homogeneous with respect to a class \(F\) of mappings if, for every \(x,y\in X\), there exists a continuous surjection \(f\in F\) such that \(f; X\to X\) and \(f(x) =y\). It is a result of Urysohn that if all points of a nondegenerate continuum are of the same order \(\kappa\), then \(\kappa\in \{2, \omega, \aleph_0, c=2^{\aleph_0}\}\). For \(\kappa=2\), such a continuum must be a simple closed curve. The author discusses the constructions of plane continua of constant order and classes of maps which either do not increase or do not decrease order. He presents several questions concerning the characterization of such continua and their generalized homogeneity properties. Among other results, it is shown that if a continuum \(X\) contains a point of order \(\omega\) and is homogeneous with respect to a class of mappings which either does not decrease or does not increase the order of points, then \(X\) is locally connected.