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Erschienen in:
13.11.2020
Computability of Products of Chainable Continua
Erschienen in: Theory of Computing Systems | Ausgabe 2/2021
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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 ai to bi 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 xi ∈{ai, bi} for some \(i\in \{1,{\dots } ,n\}\).