COMPUTABLE TYPE OF CERTAIN QUOTIENT SPACES

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.