## Completeness: When Enough is Enough

##### Doc. Math. 24, 899-914 (2019)
DOI: 10.25537/dm.2019v24.899-914

### Summary

We investigate the notion of a complete enough metric space that, while classically vacuous, in a constructive setting allows for the generalisation of many theorems to a much wider class of spaces. In doing so, this notion also brings the known body of constructive results significantly closer to that of classical mathematics. Most prominently, we generalise the Kreisel-Lacome-Shoenfield Theorem/Tseytin's Theorem on the continuity of functions in recursive mathematics.

### Mathematics Subject Classification

03F60, 03D78, 03F55

### Keywords/Phrases

constructive mathematics, computable analysis, completeness

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### Affiliation

Diener, Hannes
School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
Hendtlass, Matthew
School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand