Here we unify two results of Steinhaus and their corresponding category analogues by extending them in the settings of category bases. We further show that in any perfect translation base, every abundant Baire set contains a full subset for which our second theorem fails.
In this paper, we show that, under not too restrictive conditions, results much stronger than those obtained earlier by Hejduk can be established in category bases.
Michalski gave a short and elegant proof of a theorem of A. Kumar which states that for each set A ⊆ R, there exists a set B ⊆ A which is full in A and such that no distance between points in B is a rational number. He also proved a similar theorem for sets in R 2 . In this paper, we generalize these results in some special types of category bases.
In this paper, we first establish some equivalent formulations of non-Baire sets in category bases. We then introduce the notion of an uniform non-Baire family of sets and show that there is an uniform non-Baire family inducing a decomposition of the whole space. This phenomenon is then interpreted in the context of the famous Banah-Mazur game.
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