Remarks on the space ℵ1 in ZF

We show:(1)ℵ1 with the order topology is effectively normal, i.e., there is a function associating to every pair (A,B) of disjoint closed subsets of ℵ1 a pair (U,V) of disjoint open sets with A⊆U and B⊆V.(2)For every countable ordinal α the ordered space α is metrizable. Hence, every closed subset of α is a zero set and consequently the Čech–Stone extension of α coincides with its Wallman extension.(3)In the Feferman–Levy model where ℵ1 is singular, the ordinal space ℵ1 is base-Lindelöf but not Lindelöf.(4)The Čech–Stone extension βℵ1 of ℵ1 is compact iff its Wallman extension W(ℵ1) is compact.(5)The set L of all limit ordinals of ℵ1 is not a zero set.