Computability of Products of Chainable Continua
| Autor: | Zvonko Iljazović, Matea Čelar |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Physics
Computability 010102 general mathematics Hausdorff space Boundary (topology) 0102 computer and information sciences Type (model theory) Topological space 01 natural sciences Theoretical Computer Science Combinatorics Computational Theory and Mathematics 010201 computation theory & mathematics Unit cube Computable topological space Computable set Semicomputable set Product of chainable continua Embedding Continuum (set theory) 0101 mathematics |
| Zdroj: | Theory of Computing Systems. 65:410-427 |
| ISSN: | 1433-0490 1432-4350 |
| Popis: | 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 $i\in \{1,{\dots } ,n\}$ . |
| Databáze: | OpenAIRE |
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