Abstract
We show: (1) ℵ 1 with 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 . (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 ℵ 1 is singular, the ordinal space ℵ 1 is base-Lindelöf but not Lindelöf. (4) The Čech–Stone extension β ℵ 1 of ℵ 1 is compact iff its Wallman extension W ( ℵ 1 ) is compact. (5) The set L of all limit ordinals of ℵ 1 is not a zero set.
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