Infinitary Noetherian constructions II. Transfinite words and the regular subword topology

We show that the space of transfinite words, that is, ordinal-indexed words, over a Noetherian space is also Noetherian, under a natural topology which we call the regular subword topology. We characterize its sobrification and its specialization ordering, and we give an upper bound on its sobrification rank and on its stature.