\(\mathbb{R}\)-factorizable, simply \(sm\)-factorizable paratopological groups and their quotients
A paratopological group \(G\) is \textit{\(\mathbb{R}_i\)-factorizable}, for \(i=1,2,3,3.5\), if \(G\) is a \(T_i\)-space and for every continuous real-valued function \(f\) on \(G\), one can find a continuous homomorphism \(p:G\to H\) onto a second-countable paratopological group \(H\) satisfying the \(T_i\) separation axiom and a continuous real-valued function \(g\) on \(H\) such that \(f=g\circ p\). If we do not impose any separation restriction on \(H\), we obtain the concept of \textit{\(\mathbb R\)-factorizability}. In [\textit{L.-H. Xie} et al., Topology Appl. 160, No. 14, 1902--1917 (2013; Zbl 1285.22010)], it was proved that all concepts of \(\mathbb R\)- and \({\mathbb R}_i\)-factorizability, for \(i=1,2,3,3.5\), coincide in the class of paratopological groups. Recall that a (para)topological group \(G\) is called \textit{simply \(sm\)-factorizable} if for every co-zero set \(U\) in \(G\), one can find a continuous homomorphism \(p\) of \(G\) onto a separable metrizable (para)topological group \(H\) such that \(U=p^{-1}(p(U))\). In this paper, the authors give an affirmative answer to a problem posed by \textit{M. Sanchis} and \textit{M. Tkachenko} in [ibid. 157, No. 4, 800--808 (2010; Zbl 1185.54034), Problem 5.2]. In fact, they prove that if a paratopological group \(H\) is a continuous open homomorphic image of an \(\mathbb R\)-factorizable paratopological group, then \(H\) is \(\mathbb R\)-factorizable. They also discuss a problem posed by \textit{A. Arhangel'skii} and \textit{M. Tkachenko} [ibid. 235, 54--72 (2018; Zbl 1387.54021), Problem 7.8]: Is every continuous homomorphic image of an \(\mathbb R\)-factorizable topological group simply \(sm\)-factorizable? They give the following partial answer: If a paratopological group \(H\) is a continuous homomorphic image of an \(\mathbb R\)-factorizable paratopological group and \(H\) is weakly Lindelöf (or first-countable), then \(H\) is simply \(sm\)-factorizable. In addition, they obtain the following results: Every weakly Lindelöf totally \(\omega\)-narrow paratopological group is simply \(sm\)-factorizable. If a paratopological group \(H\) is a continuous homomorphic image of an \(\omega\)-steady \(\mathbb R\)-factorizable paratopological group, then \(H\) is \(\mathbb R\)-factorizable.
scientific article; zbMATH DE number 7047528
scientific article