## Abstract

A Tychonoff space Xp = X ∪ {p} is called a one-point extension of X if X is dense in Xp and the reminder Xp \X consists of the singleton {p}.

We study the following problem: Characterize the spaces X such that every (some) one-point extension Xp 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 Gδ-set in Xp; 2) Xp admits a local countable base at p; 3) Xp has the Fréchet-Urysohn property at p; 4) Xp 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 Gδ in Xp, for every (for some, respectively) one-point extension Xp of X. We pose several open problems for various concrete properties P.

We study the following problem: Characterize the spaces X such that every (some) one-point extension Xp 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 Gδ-set in Xp; 2) Xp admits a local countable base at p; 3) Xp has the Fréchet-Urysohn property at p; 4) Xp 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 Gδ in Xp, for every (for some, respectively) one-point extension Xp of X. We pose several open problems for various concrete properties P.

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

Journal | Topology Proceedings |

State | Published - 2022 |