Order convergence in infinite-dimensional vector lattices is not topological

Abstract

In this note, we show that the order convergence in a vector lattice $X$ is not topological unless $\dim X<\infty$. Furthermore, we show that, in atomic order continuous Banach lattices, the order convergence is topological on order intervals.

Cited by in corpus (3)

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• $um$-Topology in multi-normed vector lattices
• Unbounded Order Convergence and Universal Completions