© 1997 Copyright by Petr Simon. All rights reserved.

Answer

Lake of Wada: Answer to Problems 1 and 2 Petr Simon Asked in The Lakes of Wada Question by Stephen Watson

There is an infinite almost disjoint family A on \omega such that for every infinite subset m of \omega, the set L(m) = {a \in A: a \cap m is infinite } is either finite, or of the size continuum, never anything between.

This is an old result by Bohus Balcar and me. The proof can be found in our Boolean Handbook paper.

So denote by M the family of all infinite subsets of integers which meet infinitely many members of A. Since the size of M as well as the size of each L(m), m in M, is continuum, Kuratowski's disjoint refinement lemma applies: There is a subset D(m) of L(m) for each m in M such that D(m) is again of size continuum and D(m) and D(m') are disjoint whenever m, m' in M are distinct.

Enumerate each D(m) as { A_\alpha(m): \alpha \in c } and put U_\alpha = \cup { A_\alpha(m)^* : m \in M }.

Apparently, U_\alpha's are non-empty, open, pairwise disjoint subsets of \omega^* and the set {p \in \beta\omega: p 'subset' M } is the boundary of any union of the sets { U_\alpha : \alpha \in K } with K a non-empty subset of continuum.

Final remarks:

1. ZFC only was needed.

2. The additional demand to have the union of all U_alpha dense in \omega^* is difficult to get and I suspect it to be another equivalent of Galvin-Hechler problem.

3. Countably infinite Lake of Wada in \omega^* was done by Eric K. van Douwen when in Denton, i.e. in approximately 1985, by a quite different method; also in ZFC.