engleski

Computable intersection points

In this paper we consider a computable metric space (X, d, α), 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 K∩S contains a computable point, where S=X∖(U∪V). 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 K∩S contains a computable point under the assumption that K∩S is totally disconnected. We also prove that K∩S contains a computable point if K is a chainable continuum and S is any co-computably enumerable closed set such that K∩S 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.

Computable metric space, chainable continuum, computable compact set, computably enumerable open set, computable point

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