forcing; sigma-closed dense subset
It is consistent with the axioms of set theory that there are two co-dense partial orders, one of them $\sigma $-closed and the other one without a $\sigma $-closed dense subset.
 Vojtáš P.: Game properties of Boolean algebras
. Comment. Math. Univ. Carolin. 24 (1983), 349–369. MR 0711272