Partitioning topological spaces into countably many pieces

Abstract:

We assume $X \to (\operatorname {top}\omega + 1)_\omega ^1$ and determine which larger $\alpha$ can replace $\omega + 1$. If $X$ is first countable, any countable $\alpha$ can replace $\omega + 1$. If the character of $X$ is ${\omega _1}$, it is consistent and independent whether ${\omega ^2} + 1$ can always replace $\omega + 1$. Consistently ${\omega _1}$ cannot replace $\omega + 1$ for any $X$ of size ${\omega _1}$.