Fodor-type reflection principle and reflection of metrizability and meta-Lindelöfness
The authors introduce a new reflection principle, called Fodor-type reflection principle (FRP): Let \(\kappa\) be a cardinal of cofinality \(\geq \omega_1\). For any stationary \(S\subseteq E^\kappa_\omega\) and mapping \(g: S\to |\kappa|^{\leq\aleph_0}\), there is \(I\in|\kappa|^{\aleph_1}\) such that cf\((I)= \omega_1\); \(g(\alpha)\subseteq I\) for all \(\alpha\in I\cap S\) and for any regression \(f: I\cap S\to\kappa\) such that \(f(\alpha)\in g(\alpha)\) for all \(\alpha\in I\cap S\), there is \(\xi^*<\kappa\) such that \(f^{-1}\{\xi^*\}\) is stationary in \(\sup(I)\). This principle is strictly weaker than Fleissner's Axiom R. The authors prove that ``FRP implies that every locally separable countably tight topological space is meta-Lindelöf if all of its subspaces of cardinality \(\leq\aleph_1\) are. It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality \(\leq\aleph_1\) are. This improves a result of Balogh, who proved the same assertion under Axiom R.'' Other results in the same vein are proved in this paper, for example, it is proved that in ZFC, if \(X\) is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then \(X\) itself is also metrizable.
scientific article; zbMATH DE number 5710131