${\mathbb{R}}^{\ddot{I}{‰}_1}$-Factorizable Spaces and Groups

Abstract

A topological space $X$ is ${\mathbb{R}}^{\ddot{I}{‰}_1}$-factorizable if any continuous function $f:X\to {\mathbb{R}}^{\ddot{I}{‰}_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$ is ${\mathbb{R}}^{\ddot{I}{‰}_1}$-factorizable if and only if $X\times D(\ddot{I}{‰}_1)$, where $D(\ddot{I}{‰}_1)$ is a discrete space of cardinality $\ddot{I}{‰}_1$, is $z$-embedded in the product ${\hat{I}}^{2}X\times {\hat{I}}^{2}D(\ddot{I}{‰}_1)$ of the Stone--Cech compactifications. It is also proved that ${\mathbb{R}}^{\ddot{I}{‰}_1}$-factorizability is hereditary and countably multiplicative, that any ${\mathbb{R}}^{\ddot{I}{‰}_1}$-factorizable space is hereditarily Lindelöf and hereditarily separable, and that the existence of nonmetrizable ${\mathbb{R}}^{\ddot{I}{‰}_1}$-factorizable topological spaces and groups is independent of ZFC: under CH, all ${\mathbb{R}}^{\ddot{I}{‰}_1}$-factorizable spaces are second-countable, while under MA + $\neg$CH, the countable Fréchet--Urysohn fan is ${\mathbb{R}}^{\ddot{I}{‰}_1}$-factorizable.