Computable subcontinua of circularly chainable continua

This paper explores, in computable metric spaces, circularly chainable continua which are not chainable. Given such a continuum \(K\), if we endow it with semicomputability, its computability follows. Conditions under which semicomputability implies computability, typically topological, are extensively studied in the literature. When these conditions are not satisfied, it is natural to explore approximate approaches. In this article we investigate specific computable subcontinua of \(K\). The main result establishes that, given two points on a semicomputable, circularly chainable, but non-chainable continuum \( K \), one can approximate them by computable points such that there exists a computable subcontinuum connecting these approximations. As a consequence, given disjoint computably enumerable open sets \( U \) and \( V \) intersected by \(K\), the intersection of \( K \) with the complement of their union necessarily contains a computable point, provided that this intersection is totally disconnected.