In this paper we consider a computable metric space , a computable continuum K and disjoint computably enumerable open sets U and V in this space such that K intersects both U and V. We examine conditions under which the set contains a computable point, where . We prove that a sufficient condition for this is that K is an arc. Moreover, we consider the more general case when K is a chainable continuum and prove that contains a computable point under the assumption that is totally disconnected. We also prove that contains a computable point if K is a chainable continuum and S is any co-computably enumerable closed set such that has an isolated and decomposable connected component. Related to this, we examine semi-computable chainable continua and we get some results regarding approximations of such continua by computable subcontinua.
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