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Madeleine Birchfield (Feb 05 2024 at 16:11):Is the pseudometric space of Cauchy sequences of rational numbers Cauchy complete? If so, this implies that quotients do not preserve Cauchy completeness.
Andrew Swan (Feb 05 2024 at 16:56):Yes, that's right. The quotient is what prevents the Cauchy reals from being Cauchy complete without countable choice, because given a Cauchy sequence of Cauchy reals, there's no way to "lift" it passed the quotient map to a Cauchy sequence of Cauchy sequences.
Madeleine Birchfield (Feb 05 2024 at 17:21):(deleted)
Mike Shulman (Feb 05 2024 at 19:12):Madeleine Birchfield said:
this implies that quotients do not preserve Cauchy completeness.
...in constructive mathematics.
You really should specify when that's what you're talking about, since classical mathematics remains the default for most mathematicians.
Morgan Rogers (he/him) (Feb 05 2024 at 19:34):At this point when Madeleine or Andrew discuss something my default assumption is that they're working in constructive maths :wink:
Mike Shulman (Feb 05 2024 at 20:51):But they still should specify.