### Abstract

For a topological space X, let X
_{δ}
be the space X with the G
_{δ}
-topology of X. For an uncountable cardinal κ, we prove that the following are equivalent: (1) κ is ω1-strongly compact. (2) For every compact Hausdorff space X, the Lindelöf degree of X
_{δ}
is ≤ κ. (3) For every compact Hausdorff space X, the weak Lindelöf degree of X
_{δ}
is ≤ κ. This shows that the least ω1-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with the G
_{δ}
-topology. We also prove that the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with the G
_{δ}
-topology. For the square of a Lindelöf space, using a weak G
_{δ}
-topology, we prove that the following are consistent: (1) The least ω1-strongly compact cardinal is the supremum of the (weak) Lindelöf degrees of the squares of regular T1 Lindelöf spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular T1 Lindelöf spaces.

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

Pages (from-to) | 71-87 |

Number of pages | 17 |

Journal | Fundamenta Mathematicae |

Volume | 246 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2019 Jan 1 |

### Fingerprint

### Keywords

- And phrases: cardinal function
- G -topology
- Lindelöf degree
- ω1-strongly compact cardinal

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

_{δ}-topology and compact cardinals

*Fundamenta Mathematicae*,

*246*(1), 71-87. https://doi.org/10.4064/fm487-7-2018

**
G
_{δ}
-topology and compact cardinals
.** / Usuba, Toshimichi.

Research output: Contribution to journal › Article

_{δ}-topology and compact cardinals ',

*Fundamenta Mathematicae*, vol. 246, no. 1, pp. 71-87. https://doi.org/10.4064/fm487-7-2018

_{δ}-topology and compact cardinals Fundamenta Mathematicae. 2019 Jan 1;246(1):71-87. https://doi.org/10.4064/fm487-7-2018

}

TY - JOUR

T1 - G δ -topology and compact cardinals

AU - Usuba, Toshimichi

PY - 2019/1/1

Y1 - 2019/1/1

N2 - For a topological space X, let X δ be the space X with the G δ -topology of X. For an uncountable cardinal κ, we prove that the following are equivalent: (1) κ is ω1-strongly compact. (2) For every compact Hausdorff space X, the Lindelöf degree of X δ is ≤ κ. (3) For every compact Hausdorff space X, the weak Lindelöf degree of X δ is ≤ κ. This shows that the least ω1-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with the G δ -topology. We also prove that the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with the G δ -topology. For the square of a Lindelöf space, using a weak G δ -topology, we prove that the following are consistent: (1) The least ω1-strongly compact cardinal is the supremum of the (weak) Lindelöf degrees of the squares of regular T1 Lindelöf spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular T1 Lindelöf spaces.

AB - For a topological space X, let X δ be the space X with the G δ -topology of X. For an uncountable cardinal κ, we prove that the following are equivalent: (1) κ is ω1-strongly compact. (2) For every compact Hausdorff space X, the Lindelöf degree of X δ is ≤ κ. (3) For every compact Hausdorff space X, the weak Lindelöf degree of X δ is ≤ κ. This shows that the least ω1-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with the G δ -topology. We also prove that the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with the G δ -topology. For the square of a Lindelöf space, using a weak G δ -topology, we prove that the following are consistent: (1) The least ω1-strongly compact cardinal is the supremum of the (weak) Lindelöf degrees of the squares of regular T1 Lindelöf spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular T1 Lindelöf spaces.

KW - And phrases: cardinal function

KW - G -topology

KW - Lindelöf degree

KW - ω1-strongly compact cardinal

UR - http://www.scopus.com/inward/record.url?scp=85065483559&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065483559&partnerID=8YFLogxK

U2 - 10.4064/fm487-7-2018

DO - 10.4064/fm487-7-2018

M3 - Article

VL - 246

SP - 71

EP - 87

JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

SN - 0016-2736

IS - 1

ER -