Maximally almost periodic groups and respecting properties. In honour of Manuel López-Pellicer (Q2337493)

Let \((X,\tau)\) be a Hausdorff topological group. A compact group \(bX\) is called the Bohr compactification of \(X\) if there exists a continuous homomorphism \(i\) from \(X\) onto a dense subgroup of \(bX\) such that the pair \((bX , i)\) satisfies the universal property that if \(p \colon X\rightarrow C\) is a continuous homomorphism into a compact group \(C\), then there exists a continuous homomorphism \(j^p\colon bX\rightarrow C\) such that \(p = j^p \circ i\). The group \(X\) is called maximally almost periodic (MAP) if the group \(X^+\) is Hausdorff, where \(X^+ := (X , \tau^+)\) is the group \(X\) endowed with the Bohr topology \(\tau^+\) induced from \(bX\). A maximally almost periodic topological (MAP) group \(G\) respects \(\mathcal{P}\) if \(\mathcal{P}(G) = \mathcal{P}(G^+)\), where \(G^+\) is the group \(G\) endowed with the Bohr topology and \(\mathcal{P}\) represents the subsets of \(G\) which have the property \(\mathcal{P}\). For a Tychonoff space \(X\), \(\mathcal{B}\) denotes the family of topological properties of being a convergent sequence or a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of \(X\), respectively. The main aim of the paper is to study the relations between different respecting properties from \(\mathcal{B}\) and to show that respecting convergent sequences is the weakest one among the properties of \(\mathcal{B}\); and to characterize respecting properties from \(\mathcal{B}\) in wide classes of MAP topological groups including the class of metrizable MAP abelian groups. For the entire collection see [Zbl 1419.46002].