Bijective preimages of ω1

We study the structure of spaces admitting a continuous bijection to the space of all countable ordinals with its usual order topology. We relate regularity, zero-dimensionality and pseudonormality. We examine the effect of covering properties and ω1-compactness and show that locally compact examples have a particularly nice structure assuming MA + ¬CH. We show that various conjectures concerning normality-type properties in products can be settled (modulo set-theory) amongst such spaces.