Isometrically homogeneous and topologically homogeneous continua (Q2833071)

A continuum is a compact connected metric space. A continuum \(X\) is homogeneous if for any two elements \(x,y \in X\) there exists an autohomeomorphism \(h\) of \(X\) such that \(h(x)=y\); in the case that \(X\) admits a compatible metric \(d\) such that for all \(x,y \in X\) the homeomorphism \(h\) can be chosen to be an isometry, \(X\) is called isometrically homogeneous.NEWLINENEWLINEContinua which are topological groups and continua which are isometrically homogeneous are two particular but very important classes of continua. This paper is devoted to studying these two classes of continua and the structure of their subcontinua having nonempty interior.NEWLINENEWLINEA continuum \(X\) is: (a) indecomposable if each of its proper subcontinua has empty interior; (b) aposyndetic if for every pair of distinct points \(x,y \in X\), there exists a subcontinuum of \(X\) containing one of the points in its interior and non containing the other; (c) semi-indecomposable if for any two disjoint subcontinua of \(X\), at least one has an empty interior; and (d) mutually aposyndetic if for every pair of distinct points \(x\) and \(y\) in \(X\), there exist disjoint subcontinua \(M\) and \(N\) of \(X\) such that \(x\) is in the interior of \(M\) and \(y\) is in the interior of \(N\).NEWLINENEWLINEThe main result of this paper is: \newline THEOREM. Let \(G\) be a non-degenerate continuum which is either isometrically homogeneous or a topological group. Then, exactly one of the three following statements holds: \newline (a) \(G\) is indecomposable. \newline (b) \(G\) is semi-indecomposable and aposyndetic. \newline (c) \(G\) is mutually aposyndetic.NEWLINENEWLINEEach of the three classes (a), (b) and (c) is composed of continuum-many mutually non-homeomorphic members. Clearly, if \(M\) and \(N\) are in two different classes, then the subcontinua of \(M\) with nonempty interior play an extremely different role than the subcontinua of \(N\) with nonempty interior.NEWLINENEWLINEAll the solenoids belong to class (a), a big family of products of two solenoids belong to class (b), and class (c) is very large, in particular, contains all locally connected continua which are topological groups and another big family of products of two solenoids.NEWLINENEWLINEThe paper also contains some characterizations of solenoids and the author proves that path connected isometrically homogeneous continua are locally connected.