Computable intersection points
Zvonko Iljazović, Bojan Pažek · Computability · 2017
In this paper we consider a computable metric space [Formula: see text], 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 [Formula: see text] contains a computable point, where [Formula: see text]. 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 [Formula: see text] contains a computable point under the assumption that [Formula: see text] is totally disconnected. We also prove that [Formula: see text] contains a computable point if K is a chainable continuum and S is any co-computably enumerable closed set such that [Formula: see text] 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.