ON TWO QUESTIONS ABOUT TOPOLOGICAL (AND NONSTANDARD) EXTENSIONS

We construct, under MA, a non-Hausdorff (T 1-)topological extension ∗ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ∗ω to ∗ω. We also show (in ZFC) that for every nontrivial topological extension ∗X of a set X there exists a topology τf on ∗X, strictly finer than the Star topology, and such that (∗X, τf) is still a topological extension of X with the same function extensions ∗f. This solves two questions raised b