  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
  (i) there are 0-dimensional P_{<omega} spaces of size 2^omega,
  (ii) there are compact P_{<omega} spaces of size omega_1,
  (iii) the existence of a Psi-like examples for a compact P_{<omega} space of
size omega_1 is independent of ZFC,
  (iv) 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<=omega_1.

Logic