Extensions of functions in Mrowka-Isbell spaces

Abstract

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 extensions

(ii) for every m.a.d. family Sigma there exists a function phi:Sigma --> R that has no essential extension

and (iii) there exists a Mrowka-Isbell space Psi(Sigma) of cardinality N-1 such that every function phi:Sigma --> R with at least two different uncountable fibers, has no full extension. On the other hand, under Martin's Axiom every function phi:Sigma --> {0, 1} (respectively phi:Sigma --> R) has an essential extension if \Sigma\ < 2(N0). 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. Sigma for which every onto function phi:Sigma --> {0, 1} with infinite fibers has no essential extensions is consistent with ZFC. (C) 1997 Elsevier Science B.V. AMS classification: 54C20

54A35

04A20.