Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness

We introduce a new reflection principle which we call “Fodor-type Reflection Principle” (FRP). This principle follows from but is strictly weaker than Fleissner’s Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ≤ א2. We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ≤ א1 are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ≤ א1 are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R. Date: April 20, 2009 (22:21 JST) 2000 Mathematical Subject Classification: 03E35, 03E65, 54D20, 54D45, 54E35