On families of Lindelöf and related subspaces of $2^{\omega _1}$

Lúcia Junqueira ¹   ; Piotr Koszmider ²

¹ Departamento de Matemática Universidade de Säo Paulo Caixa Postal 66281 Säo Paulo, SP CEP: 05315-970, Brasil
² Departamento de Matemática Universidade de Säo Paulo Caixa Postal 66281 Säo Paulo, SP CEP: 05315-970, Brasil.

Fundamenta Mathematicae, Tome 169 (2001) no. 3, pp. 205-231

Résumé

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.

Détail

DOI : 10.4064/fm169-3-2

Keywords: consider families subspaces size omega omega compact zero dimensional space weight omega general which normal have lindel property closed under limits convergent omega sequences various relations among these families modulo club filter omega shown consistently possible main tools dealing subspace form cap elementary submodel size omega various results flavor obtained another tool forcing various preservation nonpreservation results topological combinatorial properties proved particular prove there may forcing which destroys lindel property compact spaces answering question juh many related questions formulated