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Sets with Steinhaus and Smital properties. (English) Zbl 1422.22007

With the basic structure, here, a topological group, sets with the Steinhaus property and with the Smital property are delineated.
Starting with a locally compact abelian group \(X\) and the completed Haar measure \(\lambda\) on \(X\), for \(A\subseteq X\) we say that \(A\) has the Steinhaus property (\(A \in \mathcal{SP}\)) if \(\operatorname{int}(A-A)\neq \phi\), the Smital property (\(A \in \mathcal{S}{m} \mathcal{P}\)) if, for any dense set \(D\), \(\lambda((A+D)^{c})=0\), and finally, the extended Smital property \((A \in \mathcal{E} \mathcal{S}m \mathcal{P})\) if, for any dense set \(D\), the set \((A+D)^{c}\) contains no set of positive Haar measure. For \(\mathcal{N}_{\lambda}\)- the \(\sigma\)-ideal of all subsets of sets of \(\mu\)-measure zero, it is shown that for \(X\), as above, \(2^{X}\setminus \mathcal{N}_{\lambda}\subset \mathcal{E} \mathcal{S}m \mathcal{P}\); if separability is added to \(X\), then it is further shown that \(\mathcal{E} \mathcal{S}m \mathcal{P}=2^{X}\setminus \mathcal{N}_{\lambda}\) and \(\mathcal{S}{m} \mathcal{P} \subset \mathcal{SP}\). It is also shown that if \(X\) is locally compact, \(\sigma\)-compact and not discrete, then \(\mathcal{S}{m} \mathcal{P} \subsetneqq \mathcal{SP}\).
The partition of \(\mathbb{R}\) by sets not belonging to \(\mathcal{S}{m} \mathcal{P}\) (or \(\mathcal{SP}\)) is studied and one such partition states that there are sets \(A, B \not \in \mathcal{SP}\) such that \(A\cup B= \mathbb{R}\).
For a proper ideal \(\mathcal{I}\) of subsets of an abelian topological group \(X\), a set \(A \subset X\) is said to have the Smital property with respect to \(\mathcal{I}\) \((A \in \mathcal{S}{m} \mathcal{P}_{\mathcal{I}})\) if \((A+D)^{c}\in \mathcal{I}\) for any dense set \(D\); it is shown that the family \(\mathcal{S}{m}\mathcal{P}_{\{\phi\}}\) consists of all subsets of \(X\) with non-empty interior.
Considering the ideal Fin consisting of finite subsets of \(X\) and \(\sigma\)-ideal Count consisting of countable subsets of \(X\), it is shown that for \(X=\mathbb{R}\), one has \(\mathcal{S}{m}\mathcal{P}_{\{ \phi\}} \subsetneqq \mathcal{S}{m}\mathcal{P}_{\mathrm{Fin}} \subsetneqq \mathcal{S}{m}\mathcal{P}_{\mathrm{Count}} \subsetneqq \mathcal{S}{m}\mathcal{P}\).
Further, in the perspective of \(\mathbb{R}\), it is shown that there is a \(G_{\delta}\)-subset of \(\mathbb{R}\) belonging to \(\mathcal{S}{m}\mathcal{P}_{\mathrm{Fin}}\setminus \mathcal{S}{m}\mathcal{P}_{\{ \phi\}}\).

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
22A05 Structure of general topological groups
54E52 Baire category, Baire spaces
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