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Every nontrivial countably productive coreflective subcategory of topological linear spaces is -productive for a large cardinal (see [10]). Unlike that case, in uniform spaces for every infinite regular cardinal , there are coreflective subcategories that are -productive and not -productive (see [8]). From certain points of view, the category of topological groups lies in between those categories above and we shall show that the corresponding results on productivity of coreflective subcategories...
Generalizing a theorem of Oxtoby, it is shown that an arbitrary product of Baire spaces which are almost locally universally Kuratowski-Ulam (in particular, have countable-in-itself π-bases) is a Baire space. Also, partially answering a question of Fleissner, it is proved that a countable box product of almost locally universally Kuratowski-Ulam Baire spaces is a Baire space.
As is well-known, every product of a locally compact space with a -space is a -space. But, the product of a separable metric space with a -space need not be a -space. In this paper, we consider conditions for products to be -spaces, and pose some related questions.
The stability of the Lindelöf property under the formation of products and of sums is investigated in ZF (= Zermelo-Fraenkel set theory without AC, the axiom of choice). It is • not surprising that countable summability of the Lindelöf property requires some weak choice principle, • highly surprising, however, that productivity of the Lindelöf property is guaranteed by a drastic failure of AC, • amusing that finite summability of the Lindelöf property takes place if either some weak choice principle...
In the present article we provide an example of two closed non--lower porous sets such that the product is lower porous. On the other hand, we prove the following: Let and be topologically complete metric spaces, let be a non--lower porous Suslin set and let be a non--porous Suslin set. Then the product is non--lower porous. We also provide a brief summary of some basic properties of lower porosity, including a simple characterization of Suslin non--lower porous sets in topologically...
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