Finite-sheeted covering spaces and a Near Local Homeomorphism Property for pseudosolenoids

Fox theory of overlays is used to characterize finite-sheeted covering spaces of pseudosolenoids (i.e. hereditarily indecomposable circle-like nonchainable continua). Let S be a P-adic pseudosolenoid, for a sequence of prime numbers P = ( p 1 , p 2 , … , p n , … ) . S admits a d-fold cover onto itself if and only if d is relatively prime to all but finitely many p i . Moreover, if C is any connected finite-sheeted covering space of S then C is homeomorphic to S. These results parallel known results for solenoids and extend results of Bellamy and Heath for the pseudocircle. In addition, it is shown that most self-maps of pseudosolenoids are finite-sheeted covering maps. Namely, if E ( S ) is the space of all surjective self-maps of S with the sup metric, then the subset C ( S ) of local self-homeomorphisms of S is a dense G δ in E ( S ) . This result, based on a theorem of Kawamura, extends the near-homeomorphism property of the pseudoarc proved independently by Lewis and Smith, and relates to a question raised by Lewis in 1984, as to which other nondegenerate continua have the near-homeomorphism property.