Semigroups of Sets Without the Baire Property In Finite Dimensional Euclidean Spaces

• Main Author: Nyagahakwa, Venuste
• Format: Others
• Language: English
• Published: Linköpings universitet, Matematik och tillämpad matematik 2015
• Subjects: Mathematics; Matematik
• Online Access: http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-116679; http://nbn-resolving.de/urn:isbn:978-91-7519-146-1

A semigroup of sets is a family of sets closed under finite unions. This thesis focuses on the search of semigroups of sets in finite dimensional Euclidean spaces Rn, n ≥ 1, which elements do not possess the Baire property, and on the study of their properties. Recall that the family of sets having the Baire property in the real line R, is a σ-algebra of sets, which includes both meager and open subsets of R. However, there are subsets of R which do not belong to the algebra. For example, each classical Vitali set on R does not have the Baire property. It has been shown by Chatyrko that the family of all finite unions of Vitali sets on the real line, as well as its natural extensions by the collection of meager sets, are (invariant under translations of R) semigroups of sets which elements do not possess the Baire property. Using analogues of Vitali sets, when the group 𝒬 of rationals in the Vitali construction is replaced by any countable dense subgroup 𝒬 of reals, (we call the sets Vitali 𝒬-selectors of R) and Chatyrko’s method, we produce semigroups of sets on R related to 𝒬, which consist of sets without the Baire property and which are invariant under translations of R. Furthermore, we study the relationship in the sense of inclusion between the semigroups related to different 𝒬. From here, we define a supersemigroup of sets based on all Vitali selectors of R. The defined supersemigroup also consists of sets without the Baire property and is invariant under translations of R. Then we extend and generalize the results from the real line to the finite-dimensional Euclidean spaces Rn, n ≥ 2, and indicate the difference between the cases n = 1 and n ≥ 2. Additionally, we show how the covering dimension can be used in defining diverse semigroups of sets without the Baire property.