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Let  be a separating family of lattice seminorms on a vector lattice , then  is called a multi-normed vector lattice (or MNVL). We write  if  for all . A net  in an MNVL  is said to be unbounded -convergent (or -convergent) to  if  for all . -Convergence generalizes -convergence (Deng et al. in Positivity 21:963–974, ; Kandić et al. in J Math Anal Appl 451:259–279, ) and -convergence (Zabeti in Positivity, . doi:), and specializes -convergence (Aydın et al. in Unbounded -convergence in lattice-normed vector lattices. ) and -convergence (Dabboorasad et al. in -Convergence in locally solid vector lattices. ). -Convergence is always topological, whose corresponding topology is called unbounded -topology (or -topology). We show that, for an -complete metrizable MNVL , the -topology is metrizable iff  has a countable topological orthogonal system. In terms of -completeness, we present a characterization of MNVLs possessing both Lebesgue’s and Levi’s properties. Then, we characterize MNVLs possessing simultaneously the -Lebesgue and -Levi properties in terms of sequential -completeness. Finally, we prove that every -bounded and -closed set is -compact iff the space is atomic and has Lebesgue’s and Levi’s properties.

um-Topology in multi-normed vector lattices