Fodor-type reflection principle and reflection of metrizability and meta-Lindelöfness
type: Item
instance of: scholarly article
title: Fodor-type reflection principle and reflection of metrizability and meta-Lindelöfness
zbMATH Open document ID: 1205.03058
author: Sakaé Fuchino
author: István Juhász
author: Lajos Soukup
author: Zoltán Szentmiklóssy
author: Toshimichi Usuba
published in: Topology and its Applications
publication date: 2010-05-19T00:00:00Z
review text: 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.
reviewed by: T. Thrivikraman
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E65
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54D45
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 5710131
zbMATH Keywords: Axiom R
zbMATH Keywords: reflection principle
zbMATH Keywords: locally compact
zbMATH Keywords: meta-Lindelöf
zbMATH Keywords: metrizable
MaRDI profile type: Publication
full work available at URL: j.topol.2009.04.068
OpenAlex ID: W1974538622
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DOI: 10.1016/J.TOPOL.2009.04.068
description: scientific article; zbMATH DE number 5710131