Chainable and Circularly Chainable Co-r.e. Sets in Computable Metric Spaces

We investigate under what conditions a co-recursively enumerable set S in a computable metric space (X, d, α) is recursive. The topological properties of S play an important role in view of this task. We first study some properties of computable metric spaces such as the effective covering property. Then we examine co-r.e. sets with disconnected complement, and finally we focus on study of chainable and circularly chainable continua which are co-r.e. as subsets of X. We prove that, under some assumptions on X, each co-r.e. circularly chainable continuum which is not chainable must be recursive. This means, for example, that each co-r.e. set in R or in the Hilbert cube which has topological type of the Warsaw circle or the dyadic solenoid must be recursive. We also prove that for each chainable continuum S which is decomposable and each ε > 0 there exists a recursive subcontinuum of S which is ε−close to S.