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In mathematics, specifically in order theory and functional analysis, a filter F {\displaystyle {\mathcal {F}}} in an order complete vector lattice X {\displaystyle X} is order convergent if it contains an order bounded subset := { x ∈ X : a ≤ x and x ≤ b } {\displaystyle :=\{x\in X:a\leq x{\text{ and }}x\leq b\}} ] and if F , {\displaystyle {\mathcal {F}},}
Order convergence plays an important role in the theory of vector lattices because the definition of order convergence does not depend on any topology.
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