Computable intersection points

Article type: Research Article

Authors: Iljazović, Zvonko^a | Pažek, Bojan^b

Affiliations: [a] Department of Mathematics, Faculty of Science, University of Zagreb, Croatia. zilj@math.hr | [b] Faculty of Architecture, University of Zagreb, Croatia. bojan.pazek@arhitekt.hr

Abstract: 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.

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

DOI: 10.3233/COM-170079

Journal: Computability, vol. 7, no. 1, pp. 57-99, 2018