## Abstract

A Tychonoff space X_{p} = X ∪ {p} is called a one-point extension of X if X is dense in X_{p} and the reminder X_{p} \X consists of the singleton {p}. We study the following problem: Characterize the spaces X such that every (some) one-point extension X_{p} of X has a given local topological property P at the point p. The list of properties P considered in the paper includes, among others: 1) {p} is a G_{δ}-set in Xp; 2) X_{p} admits a local countable base at p; 3) X_{p} has the Fréchet-Urysohn property at p; 4) X_{p} has countable tightness at p. One of our main results states that a Tychonoff space X is Lindelöf (not pseudocompact) iff the point p is of type G_{δ} in X_{p}, for every (for some, respectively) one-point extension X_{p} of X. We pose several open problems for various concrete properties P.

Original language | English |
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Pages (from-to) | 195-208 |

Number of pages | 14 |

Journal | Topology Proceedings |

Volume | 59 |

Early online date | 8 May 2021 |

State | Published - 1 Jan 2022 |

## Keywords

- Fréchet-Urysohn property
- G-set
- Lindelöf space
- One-point extension
- Stone-Čech compactification
- character
- zero-set

## ASJC Scopus subject areas

- Geometry and Topology