The generalizations of first countable spaces (Q809390)

The author introduces \(w_{\kappa}\)-spaces for any cardinal \(\kappa\). When \(\kappa =1,\omega_ 1,\infty\), the spaces are respectively Fréchet spaces, w-spaces in the sense of \textit{G. Gruenhage} and first countable spaces. He gives a characterization of \(w_{\kappa}\)-spaces as the images of metrizable spaces by \(w_{\kappa}\)-maps. He constructs a model in which there is a countable space with character \(\omega_ 1\) which is a \(w_{\kappa_ 1}\)-space but not \(w_{\kappa_ 2}\)-space for any cardinal \(\kappa_ 1<\kappa_ 2\).