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.