Factorization of Bijections onto Ordered Spaces

We identify a class of subspaces of ordered spaces $\mathcal L$ for which the following statement holds: If $f:X\to L\in \mathcal L$ is a continuous bijections of a zero-dimensional space $X$, then $f$ can be re-routed via a zero-dimensional subspace of an ordered space that has weight not exceeding that of $L$.