Reflecting point-countable open families
by
Zoltan Balogh
Miami University, Oxford,OH

Theorem If X is a space with density no bigger than \omega₁ such that every subspace of size at most \omega₁ has a point-countable base, then X has a point-countable base.

Corollary (Dow). If X is countably compact with every subspace of size \omega₁ metrizable, then X is metrizable. Moreover, countably compact can be replaced by any closed hereditary property P such that P+point-countable base implies second countable.

Theorem. If every subspace of size \omega₁ of a space X with density no bigger than \omega₁ and tightness \omega is meta-Lindelöf, then X is hereditarily meta-Lindelöf.

Theorem (Axiom R). If every subspace of size no bigger than \omega₁ of a locally compact space X has a point-countable base, then X is metrizable.

Date received: February 1, 2001