Baire irresolvable spaces and lifting for a layered ideal

Abstract

We show the consistency (modulo reasonable large cardinals) of the existence of a topological space of power [aleph]1 with no isolated points such that any real values function on it has a point of continuity. This is deduced from the following (by Kunen, Szymanski and Tall). We prove that if 2[lambda] = [lambda]+, l is a [lambda]-complete ideal on a regular [lambda] which is layered, then the natural homomorphism from (as Boolean algebras) can be lifted, i.e., there is a homomorphism h from ([lambda]) into itself with kernel I such that for every A [subset of or equal to] [lambda] we have [equiv] h(A) (mod l).