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- Thread starter Mr Davis 97
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Since the real numbers are both a metric space and a partially ordered set, the use of the term 'complete' is ambiguous, as each of the two possibilities gives a different definition of 'complete'. Fortunately, for the real numbers, the two definitions are logically equivalent. That's not the same thing as being equivalent by definition, as the definitions are different. But for the real numbers, satisfying one definition implies that the other is satisfied and vice versa.

More generally, for a metric space that is also a partially ordered set, I wonder if it can be proven that Cauchy completeness implies supremum completeness and/or vice versa.

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fresh_42

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More generally, for a metric space that is also a partially ordered set, I wonder if it can be proven that Cauchy completeness implies supremum completeness and/or vice versa.

The order on the space might have nothing to do with the metric, so I don't see any hope for this being true in general. For example, take ##X=[0,1]## with the usual metric and use a bijection ##f:X\to (0,1)## to induce an order ##\leq_X## on ##X## defined by ##a\leq_Xb## if ##f(a)\leq f(b)##. Then ##X## is complete in the Cauchy sense but is isomorphic as an ordered set to ##(0,1)##, so not supremum complete. Conversely, you can consider ##(0,1)## as a metric space with an order defined by a bijection ##g:(0,1)\to [0,1]## as before to get a counterexample for the reverse implication.

I think this is false. Equipping ##\mathbb{R}## with ##d_1(x,y)=|x-y|## and ##d_2(x,y)=|\arctan(x-y)|## gives homeomorphic but non-isometric spaces. For a trivial example, take the two-element discrete topological space and give it metrics which assign different distances to the pair of distinct points.On the other hand, a metric on a topological space is unique up to isometries.

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fresh_42

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Correct, I confused the order. A completion to a given metric is unique (sort of) and not the other way around. Thanks for correction.I think this is false. Equipping ##\mathbb{R}## with ##d_1(x,y)=|x-y|## and ##d_2(x,y)=|\arctan(x-y)|## gives homeomorphic but non-isometric spaces. For a trivial example, take the two-element discrete topological space and give it metrics which assign different distances to the pair of distinct points.

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