Let $(X,\leq)$ be an ordered set with countable infima and countable suprema. We can define $\limsup$ and $\liminf$ in the obvious fashion and we say a sequence converges if they coincide.
So we get a notion of convergence and limit. This is backwards from the way one usually introduces limits.
Under which reasonable circumstances can we now get (possibly a generalization of) a metric yielding the same notion of convergence and limits?