Point-free topological spaces, functions and recursive points; filter foundation for recursive analysis. I. (English) Zbl 0946.03052
The authors build on earlier ideas [I. Kalantari and G. Weitkamp, Ann. Pure Appl. Log. 29, 1-27 (1985; Zbl 0569.03018)].
Author’s abstract: We develop a point-free approach to the study of topological spaces and functions on them, establish platforms for both and present some findings on recursive points. (The effectivization of the functions on our spaces and related results are presented in a sequel.) In the first sections of the paper, we obtain conditions under which our approach leads to the generation of ideal objects (points) with which mathematicians work. Next, we apply the effective version of our approach to the real numbers, and make exact connections to the classical approach to recursive reals. In the succeeding sections of the paper, we introduce machinery to produce functions on topological spaces and find succinct conditions which will be effectivized in our sequel.
Author’s abstract: We develop a point-free approach to the study of topological spaces and functions on them, establish platforms for both and present some findings on recursive points. (The effectivization of the functions on our spaces and related results are presented in a sequel.) In the first sections of the paper, we obtain conditions under which our approach leads to the generation of ideal objects (points) with which mathematicians work. Next, we apply the effective version of our approach to the real numbers, and make exact connections to the classical approach to recursive reals. In the succeeding sections of the paper, we introduce machinery to produce functions on topological spaces and find succinct conditions which will be effectivized in our sequel.
Reviewer: R.Downey (Wellington)
MSC:
| 03D45 | Theory of numerations, effectively presented structures |
| 03C57 | Computable structure theory, computable model theory |
| 03D80 | Applications of computability and recursion theory |
| 54A20 | Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) |
Keywords:
computability; recursive analysis; computable analysis; recursive topology; computable topology; point-free topology; continuity; recursive points; recursive realsCitations:
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