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In Wikipedia:

A subset $S$ of $\mathbb{R}^n$ is bounded with respect to the Euclidean distance if and only if it bounded as subset of $\mathbb{R}^n$ with the product order.

More generally, I was wondering for a set which is both a metric space and partially-ordered set, when the boundedness of a subset wrt the metric and wrt the order agree, or just one implies the other not the other way around?

Thanks and regards!

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    Certainly if your metric is bounded you're going to have problems.2011-04-19

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For a functional analyst, the natural setting for questions of this sort is the theory of Banach lattices, i.e., vector spaces with both a norm and a lattice structure. The typical examples are the $L^p$-spaces and these are very useful for looking at the kind of relationships between the two structures that seem to interest you. Particularly suggestive are the two extreme cases $p=1$ (where the norm is order continuous) and $p=\infty$ (where the property you mention---equivalence of boundedness in the topological and in the order sense---is valid).