Ja n 19 97 A universal Polish G - space

If G is a Polish group, then the category of Polish G-spaces and continuous G-mappings has a universal object. §0. Preface It is known from [2] that 0.1. Theorem (de Vries). Let G be a separable locally compact metric group. Then the category of continuous G-maps and separable complete metric G-spaces has a universal object. In other words, there is a separable complete metric space X on which G acts continuously, and such that for any other separable completely metrizable Y on which there is a continuous G action we can find a continuous π : X → Y such that for all x0, x1 ∈ X, g ∈ G g · x0 = x1 ⇔ g · π(x0) = π(x1). On the other hand in [1]: 0.2. Theorem (Becker, Kechris). Let G be a Polish group – that is to say a separable topological group admitting a complete metric. Then the category of Borel G-maps and separable complete metric G-spaces has a universal object. While here we unify these with: 0.3. Theorem. Let G be a Polish group. Then the category of continuous G-maps and separable complete metric G-spaces has a universal object. The proof gives incidental information beyond the original goal. We may relax the condition that G be Polish to simply being a separable metric group, and even in this case one has a Polish G-space that is universal for all separable metric G-spaces: 0.4. Theorem. Let G be a separable metric group. Then there is Polish X such that: (i) G acts continuously on X; and whenever G acts continuously on some separable metric space Y we may find π : Y → X so that ∗Research partially supported by NSF grant DMS 96-22977 1