Extensions of functions in Mrowka-Isbell spaces

For an almost disjoint family (a.d.f.) Sigma of subsets of omega, let Psi(Sigma) be the Mrowka-Isbell space on Sigma. In this article we will analyze the following problem: given an a.d.f. Sigma and a function phi:Sigma --> {0: 1} (respectively phi:Sigma --> R) is it possible to extend phi continuously to a big enough subspace Sigma boolean OR N of Psi(Sigma) for which cl(Psi(Sigma)) N superset of Sigma? Such an extension is called essential. We will prove that: (i) for every a.d.f. Sigma of cardinality 2(N0) we can find a function phi:Sigma --> {0, I} without essential extension