Remarks on the space in ZF

We show:(1) ith the order topology is effectively normal, i.e., there is a function associating to every pair ( A , B ) of disjoint closed subsets of ℵ 1 a pair ( U , V ) of disjoint open sets with A ⊆ U and B ⊆ V . ery 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. Feferman–Levy model where ℵ 1 is singular, the ordinal space ℵ 1 is base-Lindelöf but not Lindelöf. ch–Stone extension β ℵ 1 of ℵ 1 is compact iff its Wallman extension W ( ℵ 1 ) is compact. t L of all limit ordinals of ℵ 1 is not a zero set.