Can $\omega_1$ be represented as union of an uncountable collection of cofinal, pairwise disjoint subsets? Can $\omega_1$ (the first uncountable ordinal) be represented as union of an uncountable collection of cofinal, pairwise disjoint subsets?

For an explicit example of an uncountable partition of $\omega_1$ into cofinal sets, for each $\alpha<\omega_1$ let $A_\alpha = \{ \beta<\omega_1\mid \exists \gamma<\omega_1,\thinspace \beta=\gamma +\omega^\alpha\}$. That is, partition $\omega_1$ by looking at the last summand in the Cantor normal form of each element of $\omega_1$.
