Hereditarily \(\mathbb R\)-factorizable groups
The aim of this paper is to indicate classes of hereditarily \(\mathbb R\)-factorizable topological groups and paratopological groups. A Lindelöf \(\Sigma\)-space is a continuous image of a Hausdorff space which admits a perfect mapping onto a separable metrizable space. A topological group \(G\) has the property \(LS(\mathfrak{c})\) if for every \(A\subseteq G\) with \(| A| \leq\mathfrak{c}\) there exists a Lindelöf \(\Sigma\)-space \(Y\) such that \(A\subseteq Y\subseteq G\). The main result of the paper is (Theorem 3.2): Every topological group with property \(LS(\mathfrak{c})\) is hereditarily \(\mathbb{R}\)-factorizable and is an Efimov space, i.e. for every family \(\gamma\) of \(G_\delta\)-sets in \(G\), the closure of \(\cup\gamma\) is a zero-set in \(G\). A similar result is true for paratopological groups (Theorem 3.1). Another class of hereditarily \(\mathbb{R}\)-factorizable is given by Theorem 3.6: Let \(\mathcal C\) be the minimal class of topological groups that contains all Lindelöf \(\Sigma\)-groups and is closed under taking continuous homomorphic images, subgroups, countable products, and forming \(\sigma\)-products. Then every group in this class has countable cellularity, is hereditarily \(\mathbb{R}\)-factorizable and perfectly \(k\)-normal. The paper is finished by four open problems.
\(\mathbb R\)-factorizable topological group
scientific article; zbMATH DE number 5710146
scientific article