On the Levi and Lebesgue properties (Q935941)

A topological Riesz space (TRS) \(E\) is said to have the Levi property if every increasing topologically bounded net of positive elements has a supremum. A TRS \(E(\tau)\) is said to have the Lebesgue property if \(x_\alpha \downarrow 0 \) in \(E\) implies \( x_\alpha \rightarrow 0 \) in \(\tau \). The aim of the paper is to investigate quotients of TRS's satisfying the Levi and Lebesgue properties. In the first part, permanence of the Levi property is studied for \(\lambda\)-sums of Banach lattices. It is shown that the Levi property is not an invariant of linear homeomorphisms. In general, quotients of TRS's with the Levi property may not have the Levi property. It is shown that if \(E(\tau)\) is a metrizable TRS with the \(\sigma\)-Levi property then \(E/F\) has the \(\sigma\)-Levi property for every \(\sigma\)-ideal \(F\) of \(E\). In the last part of the paper, the three space property is investigated for the Levi and Lebesgue properties. The paper contains a host of interesting and illuminating examples.