um-Topology in multi-normed vector lattices

Abstract

Let ${\mathcal {M}}=\{m_\lambda \}_{\lambda \in \Lambda }$ be a separating family of lattice seminorms on a vector lattice X, then $(X,{\mathcal {M}})$ is called a multi-normed vector lattice (or MNVL). We write $x_\alpha \xrightarrow {\mathrm {m}} x$ if $m_\lambda (x_\alpha -x)\rightarrow 0$ for all $\lambda \in \Lambda$. A net $x_\alpha$ in an MNVL $X=(X,{\mathcal {M}})$ is said to be unbounded m-convergent (or um-convergent) to x if $|x_\alpha -x |\wedge u \xrightarrow {\mathrm {m}} 0$ for all $u\in X_+$. 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 $u\tau$-convergence (Dabboorasad et al. in $u\tau$-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 $(X,{\mathcal {M}})$, 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 $\sigma$-Lebesgue and $\sigma$-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.