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We say that a real normed lattice is quasi-Baire if the intersection of each sequence of monotonic open dense sets is dense. An example of a Baire-convex space, due to M. Valdivia, which is not quasi-Baire is given. We obtain that is a quasi-Baire space iff , is a pairwise Baire bitopological space, where , is a quasi-uniformity that determines, in . Nachbin’s sense, the topological ordered space .
We give a proof of a theorem of Maćkowiak on the existence of universal n-dimensional hereditarily indecomposable continua, based on the Baire-category method.
Let G be a locally compact group with a fixed left Haar measure. Given Young functions φ and ψ, we consider the Orlicz spaces and on a non-unimodular group G, and, among other things, we prove that under mild conditions on φ and ψ, the set is well defined on G is σ-c-lower porous in . This answers a question raised by Głąb and Strobin in 2010 in a more general setting of Orlicz spaces. We also prove a similar result for non-compact locally compact groups.
We investigate an algebraic notion of decidability which allows a uniform investigation of a large class of notions of forcing. Among other things, we show how to build σ-fields of sets connected with Laver and Miller notions of forcing and we show that these σ-fields are closed under the Suslin operation.
For a non-isolated point of a topological space let be the smallest cardinality of a family of infinite subsets of such that each neighborhood of contains a set . We prove that
(a) each infinite compact Hausdorff space contains a non-isolated point with ;
(b) for each point with there is an injective sequence in that -converges to for some meager filter on ;
(c) if a functionally Hausdorff space contains an -convergent injective sequence for some meager filter...
Answering a question of Telgársky in the negative, it is shown that there is a space which is β-favorable in the strong Choquet game, but all of its nonempty -subspaces are of the second category in themselves.
We call a subset S of a topological vector space V linearly Borel if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V. It is shown that a Hamel basis of an infinite-dimensional Banach space can never be linearly Borel. This answers a question of Anatoliĭ Plichko.
In this note we show the following theorem: “Let be an almost -discrete space, where is a regular cardinal. Then is -Baire iff it is a -Baire space and every point- open cover of such that is locally- at a dense set of points.” For we obtain a well-known characterization of Baire spaces. The case is also discussed.
We consider the question of preservation of Baire and weakly Baire category under images and preimages of certain kind of functions. It is known that Baire category is preserved under image of quasi-continuous feebly open surjections. In order to extend this result, we introduce a strictly larger class of quasi-continuous functions, i.e. the class of quasi-interior continuous functions. We show that Baire and weakly Baire categories are preserved under image of feebly open quasi-interior continuous...
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