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Intersection Properties of Open Sets. II e
Author(s) -
JUHÁSZ I.,
NAGY ZS,
SOUKUP L.,
SZENTMIKLÓSSY Z.
Publication year - 1996
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1996.tb36806.x
Subject(s) - uncountable set , intersection (aeronautics) , cardinality (data modeling) , countable set , mathematics , space (punctuation) , combinatorics , second countable space , topological space , discrete mathematics , computer science , geography , cartography , data mining , operating system
A topological space is called P 2 ( P 3 , 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 2 ω , (ii) there are compact P <ω spaces of size ω 1 , (iii) the existence of a Ψ‐like examples for (ii) is independent of ZFC, (iv) it is consistent that 2 ω is as large as you wish but every first countable (and so every compact) P 2 space has cardinality ≤ω 1 .