Computability of Products of Chainable Continua

Abstract

We examine conditions under which a semicomputable set in a computable topological space is computable. In particular, we examine topological pairs (A, B) with the following property: 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. Such pairs (A, B) are said to have computable type. It is known that $(\mathcal {K},\{a,b\})$ has computable type if $\mathcal {K}$ is a Hausdorff continuum chainable from a to b. It is also known that (Iⁿ, ∂Iⁿ) has computable type, where Iⁿ is the n-dimensional unit cube and ∂Iⁿ is its boundary in $\mathbb {R}^{n}$. We generalize these results by proving the following: if $\mathcal {K}_{i}$ is a nontrivial Hausdorff continuum chainable from a_i to b_i for $i\in \{1,{\dots } ,n\}$, then $({\prod }_{i=1}^{n} \mathcal {K}_{i} ,B)$ has computable type, where B is the set of all $(x_{1} ,{\dots } ,x_{n})\in {\prod }_{i=1}^{n} \mathcal {K}_{i}$ such that x_i ∈{a_i, b_i} for some $i\in \{1,{\dots } ,n\}$.