A new short proof for the uniqueness of the universal minimal space

We give a new short proof for the uniqueness of the universal minimal space. The proof holds for the uniqueness of the universal object in every collection of topological dynamical systems closed under taking projective limits and possessing universal objects.


Introduction
In topological dynamics one studies jointly continuous actions G × X → X of (Hausdorff) topological groups G on nonempty (Hausdorff) compact spaces X. Such an action is called a topological dynamical system, and we call X a G-space. A G-space X is said to be minimal if X and ∅ are the only G-invariant closed subsets of X. By Zorn's lemma each G-space contains a minimal G-subspace. These minimal objects are in some sense the most basic ones in the category of G-spaces. For various topological groups G they have been the object of extensive study. Given a topological group G, one is naturally interested in describing all of the minimal G-spaces up to isomorphism. Such a description is given by the following construction: one can show that there exists a minimal G-space U G with the universal property that every minimal G-space X is a factor of U G , i.e., there is a continuous G-equivariant map from U G onto X. Any such G-space is called a universal minimal G-space, however it can be shown to be unique up to isomorphism. The existence of a universal minimal G-space is easy to demonstrate by choosing a minimal G-subspace of the product over all minimal G-spaces (one representative from each isomorphism class -the collection of isomorphism classes of minimal G-spaces is a set). The uniqueness turns out harder to show, since for two universal minimal G-spaces X and Y , there could be more than one epimorphism from X to Y , where an epimorphism is a surjective G-equivariant continuous map. An easy observation is that it suffices to show that a universal minimal G-space X is coalescent, i.e. every epimorphism φ : X → X is an isomorphism. If M 1 and M 2 are universal minimal G-spaces then by universality we have epimorphisms φ 1 : M 1 → M 2 and φ 2 : M 2 → M 1 . If in addition M 1 is coalescent, then φ 2 • φ 1 must be an isomorphism, and hence φ 1 and φ 2 are isomorphisms.
In the literature two different approaches for the proof of uniqueness have been offered. The algebraic approach using Ellis semigroups was laid out by Ellis in [E69] (he only treated the case of discrete G but the proof works for any G). The details can be found in [dV93] IV(4.30) and [U00] Appendix 3. The main tool is the theorem that any minimal subsystem of an enveloping semigroup (see [dV93] IV(3.5(1)) is coalescent. This theorem is proven by using the , [N52]): every non-empty compact Hausdorff right semitopological semigroup contains an idempotent. Another proof was given by Auslander in [A88]. The main idea is to show that given a minimal system (G, X) there is a cardinal κ such that (G, X κ ) contains a minimal coalescent subsystem. The main tool is the so called almost periodic sets.
We will give a proof for the uniqueness of the universal minimal space that holds in a more general setting. Given a collection K of G-spaces, we call M ∈ K, K-universal if any N ∈ K is a G-factor of M . We show that if K is closed under taking projective limits and posses universal spaces then it has a unique universal space (up to isomorphism). In particular the reader will be able to verify easily, that our result also applies to the universal equicontinuous minimal space and the universal distal minimal space.
Acknowledgements. H. Li was partially supported by NSF grants DMS-0701414 and DMS-1001625. We are grateful to Eli Glasner for helpful discussions. We thank the referee for useful comments.

Uniqueness
Theorem 1. Let G be a topological group and K a collection of G-spaces. Assume that K is closed under taking projective limits. Let M 1 , M 2 be Kuniversal spaces. Then M 1 and M 2 are isomorphic.