On families of Lindelöf and related subspaces of $2^{\omega _1}$
Volume 169 / 2001
Abstract
We consider the families of all subspaces of size $\omega _1$ of $2^{\omega _1}$ (or of a compact zero-dimensional space $X$ of weight $\omega _1$ in general) which are normal, have the Lindelöf property or are closed under limits of convergent $\omega _1$-sequences. Various relations among these families modulo the club filter in $[X]^{\omega _1}$ are shown to be consistently possible. One of the main tools is dealing with a subspace of the form $X\cap M$ for an elementary submodel $M$ of size $\omega _1$. Various results with this flavor are obtained. Another tool used is forcing and in this case various preservation or nonpreservation results of topological and combinatorial properties are proved. In particular we prove that there may be no c.c.c. forcing which destroys the Lindelöf property of compact spaces, answering a question of Juhász. Many related questions are formulated.
