YT MORE ON CARDINALITY BOUNDS INVOLVING THE WEAK LINDELOF DEGREE

We give several new bounds for the cardinality of a Hausdorff topological space X involving the weak Lindelof degree omega L(X). In particular, we show that if X is extremally disconnected, then |X| <= 2(omega L(X)pi chi(X)psi(X)), and if X is additionally power homogeneous, then vertical bar X vertical bar <= 2(omega L(X)pi chi(X)). We also prove that if X is a star-DCCC space with a G(delta) -diagonal of rank 3, then vertical bar X vertical bar <= 2(aleph 0) ; and if X is any normal star-DCCC space with a G(delta) -diagonal of rank 2, then vertical bar X vertical bar <= 2(aleph 0). Several improvements of results in [10] are also given. We show that if X is locally compact, then vertical bar X vertical bar <= omega L(X)(psi(X)) and that vertical bar X vertical bar <= omega L(X)(t(X)) if X is additionally power homogeneous. We also prove that vertical bar X vertical bar <= 2(psi c) ((X)t(X) omega L(X)) for any space with a pi-base whose elements have compact closures and that the stronger inequality vertical bar X vertical bar <=omega L(X)(psi c) ((X)t(X)) is true when X is locally H-closed or locally Lindelof.