Journal of Symbolic Logic 66 (3):1157-1185 (2001)

Abstract
In examples like the total recursive functions or the computable real numbers the canonical indexings are only partial maps. It is even impossible in these cases to find an equivalent total numbering. We consider effectively given topological T 0 -spaces and study the problem in which cases the canonical numberings of such spaces can be totalized, i.e., have an equivalent total indexing. Moreover, we show under very natural assumptions that such spaces can effectively and effectively homeomorphically be embedded into a totally indexed algebraic partial order that is closed under the operation of taking least upper bounds of enumerable directed subsets
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DOI 10.2307/2695099
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On Computable Numbers, with an Application to the N Tscheidungsproblem.Alan Turing - 1936 - Proceedings of the London Mathematical Society 42 (1):230-265.
Classical Recursion Theory.Peter G. Hinman - 2001 - Bulletin of Symbolic Logic 7 (1):71-73.
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Theorie der Numerierungen II.J. U. L. Eršov - 1975 - Mathematical Logic Quarterly 21 (1):473-584.

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