### Abstract

By analyzing Dow's construction, we introduce a general construction of regular Lindelof spaces with points Gδ. Using this construction, we prove the following: Suppose that either (1) there exists a regular Lindel of P-space of pseudocharacter ≤ ω1 and of size > 2ω, (2) CH and (ω2) hold, or (3) CH holds and there exists a Kurepa tree. Then there exists a regular Lindel of space with points Gδ and of size > 2ω. This shows that, under CH, the non-existence of such a Lindel of space has a large cardinal strength. We also prove that every c.c.c. forcing adding a new real creates a regular Lindel of space with points Gδ and of size at least (2ω1 )V.

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

Pages (from-to) | 249-260 |

Number of pages | 12 |

Journal | Fundamenta Mathematicae |

Volume | 237 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2017 |

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### Keywords

- Kurepa tree
- Lindel of space
- P-space
- Points Gδ
- Square principle

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Fundamenta Mathematicae*,

*237*(3), 249-260. https://doi.org/10.4064/fm296-8-2016

**Large regular Lindelöf spaces with points Gδ.** / Usuba, Toshimichi.

Research output: Contribution to journal › Article

*Fundamenta Mathematicae*, vol. 237, no. 3, pp. 249-260. https://doi.org/10.4064/fm296-8-2016

}

TY - JOUR

T1 - Large regular Lindelöf spaces with points Gδ

AU - Usuba, Toshimichi

PY - 2017

Y1 - 2017

N2 - By analyzing Dow's construction, we introduce a general construction of regular Lindelof spaces with points Gδ. Using this construction, we prove the following: Suppose that either (1) there exists a regular Lindel of P-space of pseudocharacter ≤ ω1 and of size > 2ω, (2) CH and (ω2) hold, or (3) CH holds and there exists a Kurepa tree. Then there exists a regular Lindel of space with points Gδ and of size > 2ω. This shows that, under CH, the non-existence of such a Lindel of space has a large cardinal strength. We also prove that every c.c.c. forcing adding a new real creates a regular Lindel of space with points Gδ and of size at least (2ω1 )V.

AB - By analyzing Dow's construction, we introduce a general construction of regular Lindelof spaces with points Gδ. Using this construction, we prove the following: Suppose that either (1) there exists a regular Lindel of P-space of pseudocharacter ≤ ω1 and of size > 2ω, (2) CH and (ω2) hold, or (3) CH holds and there exists a Kurepa tree. Then there exists a regular Lindel of space with points Gδ and of size > 2ω. This shows that, under CH, the non-existence of such a Lindel of space has a large cardinal strength. We also prove that every c.c.c. forcing adding a new real creates a regular Lindel of space with points Gδ and of size at least (2ω1 )V.

KW - Kurepa tree

KW - Lindel of space

KW - P-space

KW - Points Gδ

KW - Square principle

UR - http://www.scopus.com/inward/record.url?scp=85017433532&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017433532&partnerID=8YFLogxK

U2 - 10.4064/fm296-8-2016

DO - 10.4064/fm296-8-2016

M3 - Article

AN - SCOPUS:85017433532

VL - 237

SP - 249

EP - 260

JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

SN - 0016-2736

IS - 3

ER -