Two step iteration of almost disjoint families
We prove the following results about the (class of) spaces we call two
step iterations of \psi, where the first step is the well-known space
built from an infinite maximal almost disjoint family of infinite subsets
of the natural numbers (also called N union R). The problem is to
determine which of these spaces satisfies the Urysohn separation axiom
(any two points can be separated by open sets with disjoint closures). A
number of small uncountable cardinals are used in the proofs.
Theorem 1. There exists a maximal A_0 such that for every maximal
A_1, the iteration of A_0 followed by A_1 is not
Urysohn.
Theorem 2. Assume \mathfrak d \leq \mathfrak a. For every maximal A_0
there exist maximal A_1 such that
the iteration of A_0 followed by A_1 is not
Urysohn.
Theorem 3. Assume \mathfrak h =\mathfrak c. There exists maximal A_0
and A_1 such that the iteration of A_0 followed by A_1 is
Urysohn.