In this work, we study the computability of topological graphs, which are obtained by gluing arcs and rays together at their endpoints. We prove that every semicomputable graph in a computable metric space can be approximated, with arbitrary precision, by its computable subgraph with computable endpoints.

Abstract We examine topological pairs $(A,B)$ which have computable type, which means that the following holds: if X is a computable topological space and $f:A\rightarrow X$ is an embedding such that $f(A)$ and $f(B)$ are semicomputable sets in X , then $f(A)$ is a computable set in X . If $(A,\emptyset )$ has computable type, we say that A has computable type. In general, if a topological pair $(A,B)$ is such that the quotient space $A/B$ has computable type, then $(A,B)$ need not have computable type. We prove the following: if $A/B$ has computable type and the interior of B in A is empty, then $(A,B)$ has computable type. On the other hand, if $(A,B)$ has computable type, then $A/B$ need not have computable type even if $\mathop {\mathrm {Int}}_{A}B=\emptyset $ . Related to this, we introduce the notion of a local computable type. We show that $\mathbb {R}^{n} /K$ has local computable type if K is a compact subspace of $\mathbb {R}^{n} $ such that $\mathbb {R}^{n} \setminus K$ has finitely many connected components.

Abstract We examine conditions under which a semicomputable set in a computable topological space is computable. In particular, we examine topological spaces $\varDelta $ that have computable type, which means that any semicomputable set homeomorphic to $\varDelta $ is computable. It is known that each compact manifold has computable type. In this paper, we examine compact manifolds $M$ and $N$ and a space $M\cup _{\gamma }N$ obtained by gluing $M$ and $N$ together by way of a homeomorphism $\gamma :A\rightarrow B$, where $A$ and $B$ are closed subspaces of $M$ and $N$, respectively. We show that $M\cup _{\gamma }N$ in general need not have computable type. We prove that $M\cup _{\gamma }N$ has computable type under the additional assumption that $A$ and $B$ are contained in regular submanifolds of $M$ and $N$. We also show that the same holds for a space obtained by gluing finitely many manifolds, but not for infinitely many.