Omitting the cardinality of the continuum in scattered spaces (Q1263826)

The main result of the paper is the construction of a model of set theory in which there is a Lindelöf space of cardinality \(\geq \nvDash^{\omega}\) which has neither closed nor Lindelöf subspaces of cardinality \(\nvDash^{\omega}.\) The construction is as follows: first \(\omega\) \(\nvDash\) Cohen subsets of \(\omega_{\nvDash}\) are added, with countable conditions, to a model of GCH. Then, in an Ostaszewski-type construction, the topology of the subspace of \(\nvDash^{\omega}\nvDash\) consisting of these subsets is refined to a scattered Lindelöf topology which omits \(\omega\) \(\nvDash\). Finally, \(\nvDash^{\omega}\) is made equal to \(\omega\) \(\nvDash\) using Cohen forcing; the important properties of the space are preserved under this forcing. The authors add some remarks and examples concerning the preservation of various topological properties under forcing.