I. Juhász, Zs. Nagy, L. Soukup, Z. Szentmiklóssy:
Intersection properties of open sets, II.
A topological space is called P_2 (P_3, P_<omega) if and only if
it does not contain two (three, finitely many)
uncountable open sets with empty intersection.
We show that
 there are 0dimensional P_<{\omega} spaces of size
2^omega,
 there are compact P_<\omega spaces of size omega_1,

the existence of a Psilike examples for a compact P_<\omega space
of size omega_1 is independent of ZFC,
 it is consistent
that 2^omega is as large as you wish but every first
countable (and so every compact) P_2 space has cardinality
\lt;=omega_1.
Downloading the paper