Intrinsic topologies on ordered structures : applications

Abstract

This thesis is divided into two main parts. The first three chapters focus on the study of order and unbounded order convergence. We examine various well-established definitions of order convergence, that have emerged over time, each associated with a corresponding topology. These notions are compared in detail, with particular attention to the conditions under which they coincide or differ. Furthermore, in the context of a semi-finite measure space, we investigate the relationship between the topologies on L∞ arising from the duality (L∞, L1), and we compare these to the order topology. Notably, we establish a condition under which the Mackey topology is strictly weaker than the order topology. Compared to order convergence, unbounded order convergence is relatively new and is generally studied on Riesz spaces. In this thesis, we explore it in the broader context of lattices. Our results show that, similar to the order topology, the unbounded order topology is independent of the definition of order convergence. In addition, we extend key properties known to hold in Riesz spaces to lattices. We prove that order continuity of unbounded order convergence is equivalent to the lattice being infinitely distributive. Moreover, we show that the O-closure and uO-closure of a sublattice coincide and form a sublattice. Furthermore, we show that the uO-adherence of an ideal is an O-closed ideal. We also examine the MacNeille completion of a sublattice Y relative to that of a lattice L, identifying two conditions under which the completion of Y embeds regularly in that of L. The last chapter is dedicated to the study of lattice uniformities. It is known that for a locally solid Riesz space (X, τ ) there exists a locally solid linear topology uτ on X such that unbounded τ -convergence coincides with uτ -convergence. This topology is the weakest locally solid linear topology that agrees with τ on all order bounded subsets. Thus, for a uniform lattice (L, U), we introduce the weakest lattice uniformity U∗ on L that coincides with U on each order bounded subset of L. We see that if U is the uniformity induced by the topology of a locally solid Riesz space (X, τ ), then the U∗ -topology coincides with uτ . We also provide answers to several questions posed in [44, 37].