Metrizable fiberings of continuous images of the lexicographic square

A Tychonoff space A is metrizably fibered if and only if there exists a continuous map F : A → B onto a metrizable space B such that for each b ∈ B , F −1 ( b ) is metrizable. We resolve a question stated by V. Tkachuk by showing that every first countable Hausdorff continuous image of the lexicographic square is metrizably fibered. We also observe that an example of S. Mardešić and P. Papić resolves a related question stated by Tkachuk.