\(\mathbb R\)-factorizable groups and subgroups of Lindelöf \(P\)-groups.
The main subject of the paper are \(P\)-groups, that is, the topological groups whose \(G_\delta\)-subsets are open. A topological group \(G\) is said to be \(\mathbb R\)-factorizable if for every continuous function \(f:G \rightarrow \mathbb R\), one can find a continuous homomorphism \(p:G\rightarrow H\) onto a second countable topological group \(H\) and a continuous function \(h:H\rightarrow \mathbb R\) such that \(f=h\circ p\). A space \(X\) is said to be pseudo-\(\omega_1\)-compact if every locally finite family of open sets in \(X\) is countable. A space \(X\) is called \(\tau\)-stable if every continuous image \(Y\) of \(X\) which admits a coarser Tychonoff topology of weight \(\leq \tau\) has network weight of no more that \(\tau\). In the paper under review it is proved that for \(P\)-groups \(\mathbb R\)-factorizability is equivalent to pseudo-\(\omega_1\)-compactness and then that it is equivalent to \(\omega\)-stability. Later it is shown that the direct products of \(\mathbb R\)-factorizable \(P\)-groups as well as continuous homomorphic images of \(\mathbb R\)-factorizable groups are \(\mathbb R\)-factorizable. The author also shows that if \(G\) is a dense subgroup of a direct product of Lindelöf \(\Sigma\)-groups, then every continuous homomorphic image of \(G\) is \(\mathbb R\)-factorizable and perfectly \(\kappa\)-normal.
scientific article; zbMATH DE number 2037250
scientific article