Computability of sets with attached arcs
Abstract
We consider topological spaces $A$ which have computable type, which means that any semicomputable set in a computable topological space which is homeomorphic to $A$ is computable. Moreover, we consider topological pairs $(A,B)$, $B\subseteq A$, which have computable type which means the following: if $S$ and $T$ are semicomputable sets in a computable topological space such that $S$ is homeomorphic to $A$ by a homeomorphism which maps $T$ to $B$, then $S$ is computable. We prove the following: if $B$ has computable type and $A$ is obtained by gluing finitely many arcs to $B$ along their endpoints, then $(A,B)$ has computable type. We also examine spaces obtained in the same way by gluing chainable continua.Keywords
computable topological space, semicomputable set, computable set, computable type, chainable continuum