Point-free topological spaces, functions and recursive points; filter foundation for recursive analysis. I

Annals of Pure and Applied Logic 93 (1-3):125-151 (1998)
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Abstract

In this paper 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. In the first sections of the paper, we obtain conditions under which our approach leads to the generation of ideal objects 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

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References found in this work

On Computable Numbers, with an Application to the Entscheidungsproblem.Alan Turing - 1936 - Proceedings of the London Mathematical Society 42 (1):230-265.
Analysis without actual infinity.Jan Mycielski - 1981 - Journal of Symbolic Logic 46 (3):625-633.
A survey of lattices of re substructures.Anil Nerode & Jeffrey Remmel - 1985 - In Anil Nerode & Richard A. Shore (eds.), Recursion theory. Providence, R.I.: American Mathematical Society. pp. 42--323.
Recursive Real Numbers.Norman Shapiro - 1955 - Journal of Symbolic Logic 20 (2):177-177.
Recursive constructions in topological spaces.Iraj Kalantari & Allen Retzlaff - 1979 - Journal of Symbolic Logic 44 (4):609-625.

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