On separability of unbounded norm topology

TL;DR

This paper investigates the topological properties of unbounded norm topology in normed lattices, providing characterizations of separability and second countability, and applying results to Banach function spaces.

Contribution

It offers new characterizations of separability and second countability of un-topology based on properties of the underlying normed lattice.

Findings

Separability of un-topology is characterized by properties of the normed lattice.

An order continuous Banach function space is separable iff it has a σ-finite carrier and is separable in local convergence topology.

Conditions for un-topology to be a normal space are addressed.

Abstract

In this paper, we continue the investigation of topological properties of unbounded norm (un-)topology in normed lattices. We characterize separability and second countability of un-topology in terms of properties of the underlying normed lattice. We apply our results to prove that an order continuous Banach function space $X$ over a semi-finite measure space is separable if and only if it has a $\sigma$-finite carrier and is separable with respect to the topology of local convergence in measure. We also address the question when a normed lattice is a normal space with respect to the un-topology.