# $um$-Topology in multi-normed vector lattices

## TL;DR

This paper introduces the unbounded m-topology in multi-normed vector lattices, characterizes its metrizability and completeness, and explores conditions for compactness and properties like Lebesgue and Levi.

## Contribution

It develops the theory of $um$-convergence and $um$-topology in MNVLs, providing new characterizations of metrizability, completeness, and compactness related to Lebesgue and Levi properties.

## Key findings

- $um$-topology is metrizable iff $X$ has a countable topological orthogonal system.
- Characterization of MNVLs with Lebesgue's and Levi's properties via $um$-completeness.
- $um$-compactness of bounded, closed sets occurs iff the space is atomic with Lebesgue's and Levi's properties.

## 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 $m_\lambda(x_\alpha-x)\to 0$ 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 $\lvert x_\alpha-x \rvert\wedge u \xrightarrow{\mathrm{m}} 0$ for all $u\in X_+$. $um$-Convergence generalizes $un$-convergence \cite{DOT,KMT} and $uaw$-convergence \cite{Zab}, and specializes $up$-convergence \cite{AEEM1} and $u\tau$-convergence \cite{DEM2}. $um$-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 any $m$-bounded and $um$-closed set is $um$-compact iff the space is atomic and has Lebesgue's and Levi's properties.