The generalizations of first countable spaces

Abstract

In this paper we consider some generalizations of first countable spaces, called $w_{\kappa}$-spaces. When $\kappa=1,$ $\omega_{1},$ $\infty$, the spaces are respectively Fréchet spaces, $w$-spaces in the sense of $G$. Gruen-hage [5] and first countable spaces. We show that the $w_{\kappa}$-spaces are the images of metric spaces under certain kind of continuous maps, called $w_{\kappa}$-maps. For any cardinals $\kappa_{1} \lt \kappa_{2}$, we construct by forcing 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.