  For a stationary set S subseteq omega_1 and a ladder system C over S, a new
type of gaps called C-Hausdorff is introduced and investigated. We describe a
forcing model of ZFC in which, for some stationary set S, for every ladder C
over S, every gap contains a subgap that is C-Hausdorff. But for every ladder E
over omega_1 setminus S there exists a gap with no subgap that is E-Hausdorff.
A new type of chain condition, called polarized chain condition, is introduced.
We prove that the iteration with finite support of polarized c.c.c posets is
again a polarized c.c.c poset.
