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## Main Question or Discussion Point

Specifically, can they be determined (up to isomorphism of ordered fields) as the smallest connected ordered field?

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- #1

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Specifically, can they be determined (up to isomorphism of ordered fields) as the smallest connected ordered field?

- #2

matt grime

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Any ordered field can be considered connected if one simply uses the trivial topology.

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HallsofIvy

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Actually, both connectedness and the Heine-Borel theorem (that all closed and bounded sets are compact) can be shown to be equivalent to mono-tone convergence, least upper bound property, and Cauchy criterion. That is, given any one, you can prove the others. (Again, assuming the Eucliden metric: d(x,y)= |x- y|.)

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Anyway, thank you both.

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