Suppose X is a vector lattice and there is a notion of convergence x α σ − → x in X. Then we can speak of an "unbounded" version of this convergence by saying thatIn the literature, the unbounded versions of the norm, order and absolute weak convergence have been studied. Here we create a general theory of unbounded convergence but with a focus on uo-convergence and those convergences deriving from locally solid topologies. We will see that, not only do the majority of recent results on unbounded norm convergence generalize, but they do so effortlessly. Not only that, but the stucture of unbounded topologies is clearer without a norm. We demonstrate this by removing metrizability, completeness, and local convexity from nearly all arguments, while at the same time making the proofs simpler and more general. We also give characterizations of minimal topologies in terms of unbounded topologies and uo-convergence.

Abstract. As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly uo-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been studied in recent papers by D. Leung, V.G. Troitsky, and the aforementioned authors. We will prove that a Banach lattice is boundedly uo-complete iff it is monotonically complete. Afterwards, we study completeness-type properties of minimal topologies; minimal topologies are exactly the Hausdorff locally solid topologies in which uo-convergence implies topological convergence.

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Given a Schauder basic sequence (x k ) in a Banach lattice, we say that (x k ) is bibasic if the expansion of every vector in [x k ] converges not only in norm, but also in order. We prove that, in this definition, order convergence may be replaced with uniform convergence, with order boundedness of the partial sums, or with norm boundedness of finite suprema of the partial sums.The results in this paper extend and unify those from the pioneering paper Order Schauder bases in Banach lattices by A. Gumenchuk, O. Karlova, and M. Popov. In particular, we are able to characterize bibasic sequences in terms of the bibasis inequality, a result they obtained under certain additional assumptions.After establishing the aforementioned characterizations of bibasic sequences, we embark on a deeper study of their properties. We show, for example, that they are independent of ambient space, stable under small perturbations, and preserved under sequentially uniformly continuous norm isomorphic embeddings. After this we consider several special kinds of bibasic sequences, including permutable sequences, i.e., sequences for which every permutation is bibasic, and absolute sequences, i.e., sequences where expansions remain convergent after we replace every term with its modulus. We provide several equivalent characterizations of absolute sequences, showing how they relate to bibases and to further modifications of the basis inequality.We further consider bibasic sequences with unique order expansions. We show that this property does generally depend on ambient space, but not for the inclusion of c 0 into ℓ ∞ . We also show that small perturbations of bibases with unique order expansions have unique order expansions, but this is not true if "bibases" is replaced with "bibasic sequences".In the final section, we consider uo-bibasic sequences, which are obtained by replacing order convergence with uo-convergence in the definition of a bibasic sequence. We show that such sequences are very common.

In 1987, I. Labuda proved a general representation theorem that, as a special case, shows that the topology of local convergence in measure is the minimal topology on Orlicz spaces and L ∞ . Minimal topologies connect with the recent, and actively studied, subject of "unbounded convergences". In fact, a Hausdorff locally solid topology τ on a vector lattice X is minimal iff it is Lebesgue and the τ and unbounded τ -topologies agree.In this paper, we study metrizability, submetrizability, and local boundedness of the unbounded topology, uτ , associated to τ on X. Regarding metrizability, we prove that if τ is a locally solid metrizable topology then uτ is metrizable iff there is a countable set A with I(A) τ = X. We prove that a minimal topology is metrizable iff X has the countable sup property and a countable order basis.In line with the idea that uo-convergence generalizes convergence almost everywhere, we prove relations between minimal topologies and uo-convergence that generalize classical relations between convergence almost everywhere and convergence in measure.