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

Abstract

We give a new short proof for the uniqueness of the univer- sal 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. G on nonempty (Hausdor) 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-spaceUG with the universal property that every minimal G-space X is a factor of UG, i.e., there is a continuous G-equivariant map from UG 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 minimalG-subspace of the product over all minimalG-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 suces to show that a universal minimal G-space X is coalescent, i.e. every epimorphism : X ! X is an isomorphism. If M1 and M2 are universal minimal G-spaces then by universality we have epimorphisms 1 : M1! M2 and 2 : M2! M1. If in addition M1 is coalescent, then 2 1 must be an isomorphism, and hence 1 and 2 are isomorphisms.