### Abstract

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 ≤א_{2}.We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ≤א_{1;} are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ≤א_{1} are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that 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 (Corollary 5.2).

Original language | English |
---|---|

Pages (from-to) | 1415-1429 |

Number of pages | 15 |

Journal | Topology and its Applications |

Volume | 157 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2010 Jun |

Externally published | Yes |

### Fingerprint

### Keywords

- Axiom R
- Locally compact
- Meta-Lindelöf
- Metrizable
- Reflection principle

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*157*(8), 1415-1429. https://doi.org/10.1016/j.topol.2009.04.068

**Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness.** / Fuchino, Sakaé; Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán; Usuba, Toshimichi.

Research output: Contribution to journal › Article

*Topology and its Applications*, vol. 157, no. 8, pp. 1415-1429. https://doi.org/10.1016/j.topol.2009.04.068

}

TY - JOUR

T1 - Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness

AU - Fuchino, Sakaé

AU - Juhász, István

AU - Soukup, Lajos

AU - Szentmiklóssy, Zoltán

AU - Usuba, Toshimichi

PY - 2010/6

Y1 - 2010/6

N2 - 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.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that 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 (Corollary 5.2).

AB - 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.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that 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 (Corollary 5.2).

KW - Axiom R

KW - Locally compact

KW - Meta-Lindelöf

KW - Metrizable

KW - Reflection principle

UR - http://www.scopus.com/inward/record.url?scp=77951665240&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951665240&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2009.04.068

DO - 10.1016/j.topol.2009.04.068

M3 - Article

VL - 157

SP - 1415

EP - 1429

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

IS - 8

ER -