Some properties of one-point extensions - Leiderman, Arkady; Tkachenko, Mikhail (Topology Proceedings, 2021)
To the memory of Phillip Zenor, a founder of Topology Proceedings 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 X_p; 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.