Some properties of one-point extensions
Keywords:
One-point extension, Stone-Cech compactification, Lindelöf space, character, Fréchet-Urysohn property, $G_\delta$-set, zero-setAbstract
To the memory of Phillip Zenor, a founder of Topology Proceedings
A Tychonoff space $X_p=X \cup\{p\}$ is called a one-point extension of $X$ if $X$ is dense in $X_p$ and the reminder $X_p \backslash X$ 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 $\mathcal{P}$ at the point $p$. The list of properties $\mathcal{P}$ considered in the paper includes, among others: 1) $\{p\}$ is a $G_\delta$-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 $G_\delta$ in $X_p$, for every (for some, respectively) one-point extension $X_p$ of $X$. We pose several open problems for various concrete properties $\mathcal{P}$.
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