Extensions of functions in Mrówka-Isbell spaces

AbstractFor an almost disjoint family (a.d.f.) ∑ of subsets of ω, let Ψ(∑) be the Mrówka-Isbell space on ∑. In this article we will analyze the following problem: given an a.d.f. ∑ and a function φ: ∑ → {0, 1} (respectively φ: ∑ → R) is it possible to extend φ continuously to a big enough subspace ∑ ∪ N of Ψ(∑) for which lΨ(∑)N ⊃ ∑? Such an extension is called essential. We will prove that: 1.(i) for every a.d.f. ∑ of cardinality 2ℵ0 we can find a function φ: ∑ → {0, 1} without essential extensions;2.(ii) for every m.a.d. family ∑ there exists a function φ: ∑ → R that has no essential extension; and3.(iii) there exists a Mrówka-Isbell space Ψ(∑) of cardinality ℵ1 such that every function φ: ϵ → R with at least two different uncountable fibers, has no full extension. On the other hand, under Martin's Axiom every function φ: ∑ → {0, 1} (respectively φ: ∑ → R) has an essential extension if ¦∑¦ < 2ℵ0. Finally, we analyze these questions under CH and by adding new Cohen reals to a ground model M showing that the existence of an uncountable a.d.f. ∑ for which every onto function φ: ∑ → {0, 1} with infinite fibers has no essential extensions is consistent with ZFC