(adsbygoogle = window.adsbygoogle || []).push({}); "Open" and "closed" relations

We know that if we have convergent sequences (x_{n}) and (y_{n}) in simply ordered metric space, then x_{n}[itex]\leq[/itex]y_{n}implies that the limits x and y have x[itex]\leq[/itex]y. Also, x_{n}<y_{n}.

My instinct on noting this is to say that "<" is an "open relation" on that metric space, and that "[itex]\leq[/itex]" is its "closure" in that this pair of relations shares a certain property of open sets and their closures in a topological sense.

More generally, I would define an "open" relation to be a relation such that, if it holds pairwise for two convergent sequences (or equivalently if iteventuallyholds pairwise), it does not necessarily hold in their limits. It is "closed" if this does imply that it holds in the limit. The "closure" of a relation R is the "strictest" (in some sense I'm not certain of) relation such that if R holds pairwise on two sequences, then the closure holds for the limits.

This seems like it would be a useful concept for evaluating limits, but a Google search didn't turn up anything with my particular terminology. Is this an actual developed topic? And if so, what it the common terminology for it?

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# Open and closed relations

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