Abstract
ABSTRACTA topological space is called P2 (P3, P<ω) if and only if it does not contain two (three, finitely many) uncountable open sets with empty intersection. We show that (i) there are 0‐dimensional P<ω spaces of size 2ω, (ii) there are compact P<ω spaces of size ω1, (iii) the existence of a Ψ‐like examples for (ii) is independent of ZFC, (iv) it is consistent that 2ω is as large as you wish but every first countable (and so every compact) P2 space has cardinality ≤ω1.
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