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I was wondering what are the general spaces that the concepts limsup and liminf can apply to? Is complete lattice one of them? Also How about metric space?

What are limsup and liminf specified with respect to? A subset? A sequence/net/filter base?

How many kinds of definitions for limsup and liminf in these various cases? Are they equivalent? If not, what are the conditions for them to be equivalent?

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The notion of a $\limsup$ of a filtered directed set makes sense. Namely, let $A$ be a filtered directed set and $x_\alpha, \alpha \in A$ be an $A$-indexed family in $\mathbb{R}$. Then one can define the $\limsup$ as the infimum of $\sup_{\beta > \alpha} x_{\beta}$ over all $\alpha$. This makes sense for the $\liminf$ as well.

One needs the ordering of the range set to define the limsup, though.

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