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The notion of -limit is extended from the case of functions with values in to the case of those with values in an arbitrary complete lattice and the problem of convergence of Pareto minima related to a convex cone is considered.
We study the behaviour of ℵ-compactness, extent and Lindelöf number in lexicographic products of linearly ordered spaces. It is seen, in particular, that for the case that all spaces are bounded all these properties behave very well when taking lexicographic products. We also give characterizations of these notions for generalized ordered spaces.
A subset of a Hausdorff space is called an H-set in if for every open family in such that there exists a countable subfamily of such that . In this paper we introduce a new cardinal function and show that for every H-set of a Hausdorff space .
We define -directedness, investigate various properties to determine whether they have this property or not, and use our results to obtain easier proofs of theorems due to Laurence and Alster concerning the existence of a Michael space, i.eȧ Lindelöf space whose product with the irrationals is not Lindelöf.
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