The category of S(α)-spaces is not cowellpowered

We introduce separation axioms S[η] depending on arbitrary order types η, such that for infinite ordinals α, S[α] = S(α) as introduced by Porter and Votaw. The order types η such that S[η] is cowellpowered are characterized. In particular S(α) is non-cowellpowered for each ordinal α > 1. This generalizes the known results for finite α (Dikranjan, Giuli and Tholen, 1989 and Schröder, 1983). In this case our construction is much simpler than those in Dikranjan, Giuli and Tholen, 1989 and Schröder, 1983.