On two questions about topological (and nonstandard) extensions (Q2505584)

A topological extension of a set \(X\) with the discrete topology is defined in this paper to be a \(T_1\) topological space \({}^*X\) containing \(X\), such that every function \(f: X \to X\) extends to a continuous function \({}^*f : {}^*X \to {}^*X\) in such a way that (1) \({}^*\) is a functor, i.e., \({}^*g \circ {}^*f = {}^*(g \circ f)\) for all \(f, g : X \to X\) and (2) every point of \({}^*X\) in the closure of the set of fixed points of \(f\) is also a fixed point of \({}^*f\). Answering a question posed by \textit{M. Di Nasso} and \textit{M. Forti} [Monatsh. Math. 144, No. 2, 89-112 (2005; Zbl 1073.54015)] the author constructs, assuming Martin's Axiom (or, more generally, that the Rudin-Keisler order on \(\beta \omega\) is not downward-directed) that there is a \(T_1\), non-Hausdorff topological extension \({}^*\omega\) such that every function from \(\omega\) to itself extends uniquely to a continuous function from \({}^*\omega\) to itself. Associated with each topological extension of \(X\) is a canonical map \(v\) from \({}^*X\) to the Stone-Čech compactification \(\beta X\), which is shown to depend only on the underlying set of \({}^*X\) and the functor \({}^*\). On the other hand, the topology on \({}^*X\) is not determined by these things. In answer to another question of Di Nasso and Forti, it is shown that any proper topological extension of a set \(X\) has a coarsest topology and a finest topology making it a topological extension of \(X\), and that if \(X\) is countable these are distinct. (In fact, in one case \({}^*X\setminus X\) is dense in itself, and in the other, it is a discrete subspace.) In an addendum it is stated that the restriction of countability will be removed in a forthcoming paper. Nonstandard extensions are only touched upon in this paper. For detailed comparison between them and topological extensions the reader is referred to the Di Nasso-Forti paper.