1/k-Homogeneous, Circle-like, Indecomposable continuum, Long line, Nonmetric solenoids, Shape

We study nonmetric analogues of Vietoris solenoids. Let ? be an ordered continuum, and let p =?p1,p2,??p?=?p1,p2,?? be a sequence of positive integers. We define a natural inverse limit space S(?,p), where the first factor space is the nonmetric ?circle? obtained by identifying the endpoints of ?, and the nth factor space, n>1, consists of p1p2?pn?1 copies of ? laid end to end in a circle. We prove that for every cardinal ??1, there is an ordered continuum ? such that S(?,p) is 1/?-homogeneous; for ?>1, ? is built from copies of the long line. Our example with ?=2 provides a nonmetric answer to a question of Neumann-Lara, Pellicer-Covarrubias and Puga from 2005, and with ?=1 provides an example of a nonmetric homogeneous circle-like indecomposable continuum. We also show that for each uncountable cardinal ? and for each fixed p, there are 2^?-many 1/?-homogeneous solenoids of the form S(?,p) as ? varies over ordered continua of weight ?. Finally, we show that for every ordered continuum ? the shape of S(?,p) depends only on the equivalence class of p for a relation similar to one used to classify the additive subgroups of the rational numbers. Consequently, for each fixed ?, as p varies, there are exactly c-many different shapes, where c=2^? (and there are also exactly that many homeomorphism types) represented by S(?,p).

We study nonmetric analogues of Vietoris solenoids. Let ? be an ordered continuum, and let p =?p1,p2,??p?=?p1,p2,?? be a sequence of positive integers. We define a natural inverse limit space S(?,p), where the first factor space is the nonmetric ?circle? obtained by identifying the endpoints of ?, and the nth factor space, n>1, consists of p1p2?pn?1 copies of ? laid end to end in a circle. We prove that for every cardinal ??1, there is an ordered continuum ? such that S(?,p) is 1/?-homogeneous; for ?>1, ? is built from copies of the long line. Our example with ?=2 provides a nonmetric answer to a question of Neumann-Lara, Pellicer-Covarrubias and Puga from 2005, and with ?=1 provides an example of a nonmetric homogeneous circle-like indecomposable continuum. We also show that for each uncountable cardinal ? and for each fixed p, there are 2^?-many 1/?-homogeneous solenoids of the form S(?,p) as ? varies over ordered continua of weight ?. Finally, we show that for every ordered continuum ? the shape of S(?,p) depends only on the equivalence class of p for a relation similar to one used to classify the additive subgroups of the rational numbers. Consequently, for each fixed ?, as p varies, there are exactly c-many different shapes, where c=2^? (and there are also exactly that many homeomorphism types) represented by S(?,p).