LAUSR

Let $${\mathcal {M}}=\{m_\lambda \}_{\lambda \in \Lambda }$$M={mλ}λ∈Λ be a separating family of lattice seminorms on a vector lattice X, then $$(X,{\mathcal {M}})$$(X,M) is called a multi-normed vector lattice (or MNVL). We write $$x_\alpha \xrightarrow {\mathrm {m}} x$$xα→mx if $$m_\lambda (x_\alpha -x)\rightarrow 0$$mλ(xα-x)→0 for all $$\lambda \in \Lambda $$λ∈Λ. A net $$x_\alpha $$xα in an MNVL $$X=(X,{\mathcal {M}})$$X=(X,M) is said to be unbounded m-convergent (or um-convergent) to x if $$|x_\alpha -x |\wedge u \xrightarrow {\mathrm {m}} 0$$|xα-x|∧u→m0 for all $$u\in X_+$$u∈X+. um-Convergence generalizes un-convergence (Deng et al. in Positivity 21:963–974, 2017; Kandić et al. in J Math Anal Appl 451:259–279, 2017) and uaw-convergence (Zabeti in Positivity, 2017. doi:10.1007/s11117-017-0524-7), and specializes up-convergence (Aydın et al. in Unbounded p-convergence in lattice-normed vector lattices. arXiv:1609.05301) and $$u\tau $$uτ-convergence (Dabboorasad et al. in $$u\tau $$uτ-Convergence in locally solid vector lattices. arXiv:1706.02006v3). um-Convergence is always topological, whose corresponding topology is called unbounded m-topology (or um-topology). We show that, for an m-complete metrizable MNVL $$(X,{\mathcal {M}})$$(X,M), the um-topology is metrizable iff X has a countable topological orthogonal system. In terms of um-completeness, we present a characterization of MNVLs possessing both Lebesgue’s and Levi’s properties. Then, we characterize MNVLs possessing simultaneously the $$\sigma $$σ-Lebesgue and $$\sigma $$σ-Levi properties in terms of sequential um-completeness. Finally, we prove that every m-bounded and um-closed set is um-compact iff the space is atomic and has Lebesgue’s and Levi’s properties.