On families of Lindelöf and related subspaces of~\(2^{\omega_1}\) (Q2759156)

Let \(X\) be a topological space of cardinality greater than \(\omega_1\). Then let: \([X]^{\omega_1}\) be the family of all subsets of \(X\) with cardinality equal to \(\omega_1\); \([X]^{\omega_1}_L\) be the family of all subspaces \(Y\in [X]^{\omega_1}\) which have the Lindelöf property; \([X]^{\omega_1}_N\) be the family of all subspaces \(Y\in [X]^{\omega_1}\) which are normal; \([X]^{\omega_1}_{\circ}\) be the family of all subspaces \(Y\in [X]^{\omega_1}\) which are closed under limits of \(\omega_1\)-sequences. NEWLINENEWLINENEWLINEThe authors investigate how large the above families can be and how they are interrelated. In particular, if \(X\) is a metrizable compact space then \([X]^{\omega_1}_L=[X]^{\omega_1}_N=[X]^{\omega_1}_{\circ}=[X]^{\omega_1}\). The main results of the paper concern the case \(X=2^{\omega_1}\). First, the authors find all inclusions between the families \([2^\omega_1]^{\omega_1}_L\), \([2^\omega_1]^{\omega_1}_N\), \([2^\omega_1]^{\omega_1}_\circ\) and \([2^\omega_1]^{\omega_1}\). Next they consider the above families in the Boolean algebra of subsets of \(2^{\omega_1}\) modulo the the club filter. They show that: (a) \(0\leq [2^{\omega_1}]_L^{\omega_1}\equiv [2^{\omega_1}]_N^{\omega_1}\leq [2^{\omega_1}]_{\circ}^{\omega_1}<1\). (b) It is consistent that \(0\equiv [2^{\omega_1}]_L^{\omega_1}\equiv [2^{\omega_1}]_N^{\omega_1}\equiv [2^{\omega_1}]_{\circ}^{\omega_1}<1\). (c) Assuming the existence of an inaccessible cardinal it is consistent that \(0\equiv [2^{\omega_1}]_L^{\omega_1}\equiv [2^{\omega_1}]_N^{\omega_1}\equiv [2^{\omega_1}]_{\circ}^{\omega_1}<1\). (d) Assuming the existence of a Mahlo cardinal it is consistent that \(0\leq [2^{\omega_1}]_L^{\omega_1}\equiv [2^{\omega_1}]_N^{\omega_1}<[2^{\omega_1}]_{\circ}^{\omega_1}<1\). The third aspect of the paper is related to the question of preservation of various properties by forcing. Here the authors show the following. (1) If a c.c.c. forcing \(P\) does not preserve the Lindelöf property of compact spaces, then there is an integer \(n\) such that \(P^n\) is not c.c.c. Thus it is consistent that all c.c.c. forcings preserve the Lindelöf property of compact spaces. (2) Under CH there is a c.c.c. forcing \(P\) which does not preserve the Lindelöf property of compact spaces and which does not add \(\omega_1\)-branches of a tree of height \(\omega_1\), and Cohen forcing preserves the property of \(P\) of not adding \(\omega_1\)-branches. (3) There is a proper forcing which does not preserve the Lindelöf property of compact spaces and which does not add \(\omega_1\)-branches. (4) Every forcing which preserves \(\omega_1\)and adds \(\omega_1\)-branches destroys the Lindelöf property of a compact space.