NEW BOUNDS ON THE CARDINALITY OF HAUSDORFF SPACES AND REGULAR SPACES

Using weaker versions of the cardinal function psi(c)(X), we derive a series of new bounds for the cardinality of Hausdorff spaces and regular spaces that do not involve psi(c)(X) nor its variants at all. For example, we show if X is regular then |X| <= 2(c(X)pi chi(X)) and |X| <= 2(c(X)pi chi(X)ot(X)), where the cardinal function ot(X), introduced by Tkachenko, has the property ot(X) <= min{t(X), c(X)}. It follows from the latter that a regular space with cellularity at most c and countable p- character has cardinality at most 2(c). For a Hausdorff space X we show |X| <= 2(d(X)pi chi(X)), |X| <= d((X)pi chi(X)ot(X)), and |X| <= 2(pi w(X)dot(X)), where dot(X) <= min{ot(X), pi chi(X)}. None of these bounds involve psi(c)(X) or psi(X). By introducing the cardinal functions w psi(c)(X) and d psi(c)(X) with the property w psi(c)(X)d psi(c)(X) <=psi(c)(X) for a Hausdorff space X, we show |X| <= (pi chi(X)c(X)w psi c(X)) if X is regular and |X| <= pi chi(X)(c(X)d psi c(X)w psi c(X)) if X is Hausdorff. This improves results of. Sapirovski.i and Sun. It is also shown that if X is Hausdorff then |X| <= 2(d(X)w psi c(X)), which appears to be new even in the case where w psi(c)(X) is replaced with psi(c)(X). Compact examples show that psi(X) cannot be replaced with d psi(c)(X)w psi(c)(X) in the bound 2(psi(X)) for the cardinality of a compact Hausdorff space X. Likewise,psi(X) cannot be replaced with d psi(c)(X)w psi(c)(X) in the Arhangelskii-Sapirovskii bound 2(L(X)t(X)psi(X)) for the cardinality of a Hausdorff space X. Finally, we make several observations concerning homogeneous spaces in this connection.