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No. The cofinality of an unbounded subset of κ is the same as the cofinality of κ. However, ωω has cofinality ω, but one of its club subsets is {α|ω0 < α < ω1} ∪ {ωβ|β < ω} which has cardinality ℵ1. JRSpriggs (talk) 01:32, 5 June 2021 (UTC)Reply