Abstract: | We construct, under MA, a non-Hausdorff (T1-)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 countable 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 by M. Di Nasso and M. Forti. |