© 1997 Copyright by Stephen Watson. All rights reserved.

Question

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

On October 11, 1995, Max Ganster asked me:

Does there exist an infinite closure-preserving cellular family in \betaN \N? (Of course, an infinite locally finite cellular family does not exist. The question is whether 'locally finite' can be weakened to 'closure preserving')

I replied on 18 Oct 1995 that there is a consistent positive answer:

Under the continuum hypothesis, there are c many open sets in \omega^* i.e. \betaN \N, called { U_\alpha: \alpha \in c } such that if we let H be the complement in \omega^* of the union of the U_\alpha's, then the boundary of each U_\alpha = H ! That is, we have a version of Lakes of Wada for \omega^*. Now these U_\alpha's are a closure preserving family since the closure of any or some or all of them always adds the same set namely H!

The construction from CH is not hard. We build a cellular family of basic open sets in \omega^* and divide them into c subfamilies. Our set U_\alpha will be the union of the \alpha-th subfamily. Now we do this by induction. We make sure that for every basic open set A in \omega^* either this lies inside finitely many basic open sets already assigned or else we put a basic open set contained in A into each of the subfamilies. This guarantees that any nbhd of H will intersect each family.

Problem 1: Does there exist, in ZFC, an infinite closure-preserving cellular family in \betaN \N ?

Now the construction works under the much weaker set-theoretic hypothesis p = h since then we just work in an induction of length h where h is the height of a tree \pi-base. This p = h follows, for example from continuum hypothesis or from Martin's axiom.

This method seems to also build a set-theoretic object whose existence in ZFC is unknown and a (famous?) open problem. Namely, we build a maximal almost disjoint family of subsets of omega such that any infinite set not in the ideal of this family meets c-many members of the family in an infinite set.

So I think I can do no better.

Problem 2: Does there exist, in ZFC, an infinite family of open sets in \betaN \N with equal boundary?

So this shows that the question of closure preserving cellular families is actually closely related to Lakes of Wada.

Max gave some background on 19 Oct 1995: In a joint paper with Nurettin Ergun and K. Dlaska entitled "Countably S-closed spaces" (Math. Slovaca 44 (1994) , 337-348 , we considered spaces such that the closure of the union of any infinite cellular family does not equal the union of the closures. Such spaces certainly have the property that every closure preserving cellular family has to be finite. But is the converse true? Ganster thought \omega^* might be a counterexample (\omega^* does not have the former property).

On 23 Oct 1995, I suggested:

Let us define the Wada number of a space X to be the supremum of all cardinals kappa such that there is a disjoint family of open sets { U_\alpha: \alpha \in \kappa} such that letting Z = X \ \cup { U_\alpha: \alpha \in \kappa}, we have, for all \alpha, the boundary of U_\alpha = Z. The classical example called lakes of Wada due to Yoneyama around 1920 showed that one can take an island with two lakes and dig canals so that all land is arbitrarily close to sea and each of the two lakes. Thus the Wada number of the plane is at least 3 ( of course he wanted his open sets to be simply connected but we don't care!).

The example above shows that this number is c for N^*. Yoneyama's proof shows that the Wada number of the plane equals omega.

The general problem is:

Calculate the Wada number of each space.

Specific problems include: How does Wada number behave under subspace, continuous image, perfect preimage, inverse limit, products etc.

Also is there any relationship between Wada number and cardinal functions? e.g. Wada number is less equal to cellularity. What is the effect if any of compactness or its variations or metrizability or its variations on Wada number?

Problem 3: Is the Wada number always attained? (i.e. the Wada number is defined to be a supremum; is this supremum always a maximum?).