Contribution
Effective properties of sets and functions in metric spaces with computability structure

https://doi.org/10.1016/S0304-3975(98)00301-6Get rights and content
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Abstract

We consider an abstract metric space with a computability structure and an effective separating set. In this article, we also introduce an effectively σ-compact space. The computability of real-valued functions on such a space is defined. It is shown that some of typical propositions in a metric space, namely Baire category theorem, Tietze's extension theorem and decomposition of unity, can be effectivized. It is also proved that computable functions are dense in continuous functions.

Keywords

Metric space
Computability structure
Effective σ-compactness
Computable function
Effective Tietze's extension theorem

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