## Abstract

A topological Hausdorff space X is sequentially linearly Lindelöf if for every uncountable regular cardinal κ≤w(X) and every A⊆X of cardinality κ there exists B⊆A of cardinality κ which converges to a point. We prove that the existence of a good (μ,λ)-scale for a singular cardinal μ of countable cofinality and a regular λ>μ implies the existence of a sequentially linearly Lindelöf space of cardinality λ and weight μ which is not Lindelöf.Corollaries of the main result are: (1) it is consistent to have linearly Lindelöf non-Lindelöf spaces below the continuum; (2) it is consistent to have a realcompact linearly Lindelöf non-Lindelöf space below 2^{אω}; (3) it is consistent to have a Dowker topology on א_{ω+1} in which every subset of cardinality א_{n}, n>0, has a converging subset of the same cardinality; (4) the nonexistence of sequentially linearly Lindelöf non-Lindelöf spaces implies the consistency of large cardinals.

Original language | English |
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Pages (from-to) | 135-144 |

Number of pages | 10 |

Journal | Topology and its Applications |

Volume | 128 |

Issue number | 2-3 |

DOIs | |

State | Published - 15 Feb 2003 |

## Keywords

- Complete accumulation
- Inner models
- Large cardinals
- Linearly Lindelöf spaces
- PCF-theory
- Realcompact spaces
- Singular cardinals
- Square principle

## ASJC Scopus subject areas

- Geometry and Topology