$$1/\kappa $$ 1 / κ -Homogeneous long solenoids

Abstract

We study nonmetric analogues of Vietoris solenoids. Let be an ordered continuum, and let p = p 1 , p 2 , . . . 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 p 1 p 2 . . . p n−1 copies of laid end to end in a circle. We prove that for every cardinal κ ≥ 1, there is an ordered continuum such that S( , p) is relation similar to one used to classify the additive subgroups of Q. Consequently, for each fixed , as p varies, there are exactly c-many different shapes, where c = 2 ℵ 0 , (and there are also exactly that many homeomorphism types) represented by S( , p).