© 1997 Copyright by Stephen Watson. All rights reserved.

Question

The Clopen Group Question Stephen Watson

Problem 1: Is there a compact Hausdorff space X such that X is homeomorphic to X \oplusX \oplusX but X is not homeomorphic to X \oplusX?

Around 1990, Jacek Nikiel asked me this question. I claimed that there was such an X and even published this claim in "Construction of Topological Spaces: Planks and Resolutions" in Recent Progress in Topology, M. Husek and J. van Mill (eds.) pp. 673-757, North Holland 1992.

Let us define a Hanf-Jonsson space to be any compact zero-dimensional space Y with infinitely many isolated points such that removing any isolated point leaves a space not homeomorphic to the original one and yet removing any two non-isolated points does leave a space which is homeomorphic to Y.

This seems counter-intuitive, I admit!

I proposed to take Y and replace each isolated point with a copy of Y, to iterate this process and then take the inverse limit. So if we remove two ïsolated points", we have really removed two copies of the space and recovered the original space and so X is homeomorphic to X \oplusX \oplusX. The problem with this plan is that I do not know how to prove X is not homeomorphic to X \oplusX.

In Japan in November 1991, Toshiji Terada presented me with a preprint which led me to consider the possibility of representing commutative semigroups by the homeomorphism types of the clopen subsets of a topological space. The questions below were composed in January 1997.

Definition: Let us define the clopen semigroup of a topological space X to be the set of equivalence classes (under homeomorphism) of the nonempty clopen subspaces of X equipped with the operation of free sum.

So, if A, B, C are clopen subspaces of X, then we say that [A] + [B] = [C] if and only if A \oplusB is homeomorphic to C.

Problem 2: For which abelian groups G is there a space X whose clopen semigroup is isomorphic to G? For which finite abelian groups? For which finitely-generated abelian groups? Can all abelian groups be so represented? Can the two point group be so represented?

Problem 3: For which abelian groups G and distinguished elements g in G is there a space X whose clopen semigroup is isomorphic to X such that [X] is identified with g? Can all abelian groups and distinguished elements be so represented?

Proposition
X has a \pi-base of clopen sets homeomorphic to X if and only if the clopen semigroup is a group.

Proof: Suppose X has a \pi-base of clopen sets homeomorphic to X. This means that, for all A, there is B such that B + X = A. In particular, this means that, for all A, B, there is D such that D + X = A and then there is E so that X = B + E. This means, letting C = D + E, we have C so that C + B = A. Now, in particular, this means that, for all A, there is C so that C + A = A. Now take an arbitrary B. Find Z so that A + Z = B. Now C + B = C + A + Z = A + Z = B and we have shown that C is the identity and, so that the clopen semigroup is a group.

Definition: A compact zero-dimensional space with a clopen group (with more than one element) is a Terada space.

Problem 4: Is there a Terada space?

I believe that Hanf-Jonsson and Terada spaces may be related.

Conjecture: If there is a Terada space with a clopen group of size 2, then X can be mapped continuously onto a Hanf-Jonsson space.

Problem 5: Is every Terada space (with a finite clopen group) continuously mappable onto a Hanf-Jonsson space?

Here is a possible approach to proving the conjecture. The space X may be identified with either element of Z₂. Let Z be the clopen set which is identified with 1 in Z₂. Take a infinite maximal disjoint family of clopen copies of Z in X and then identify each of these copies to a point. Now certainly we get a Hanf-Jonsson space except that it may fail to be Hausdorff! Can the family of clopen copies be chosen so that the quotient space is Hausdorff?