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Mathematics
Topology and Analysis
Relation of completeness to the l.u.b. property?
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[QUOTE="mathwonk, post: 5836737, member: 13785"] This was a confusing point to me, as I had heard there is only one complete ordered field, namely the real numbers, but then I saw examples of other "complete" non archimedean ordered fields. A correct statement seems to be that there is only one complete archimedean ordered field. The confusion occurs because of the two meanings of "complete". The lub version of complete also implies archimedean, whereas the Cauchy version of complete does not. So indeed there is only one lub-complete ordered field. Google some examples of complete non archimedean ordered fields. They tend to involve Laurent series. To get an example start from any non archimedean ordered field and form the Cauchy completion. This is discussed in standard old algebra books from my student days like Van der Waerden and Lang, but perhaps not in more modern ones like Dummitt and Foote. [/QUOTE]
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Relation of completeness to the l.u.b. property?
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