The equicontinuous structure relation and extension of continuous equivariant functions

Jan de Vries
1986 Rocky Mountain Journal of Mathematics  
In this paper we study injective objects in the category of all compact Hausdorff G-spaces, using methods from topological dynamics. In particular, we consider the question of when the equicontinuous structure relation of a subflow is the restriction of the equicontinuous structure relation of the full flow. Some necessary and sufficient conditions are given, one in terms of almost periodic functions on the flow, and another in terms of injective objects in the category of all compact Hausdorff
more » ... l compact Hausdorff G-spaces. 1980 Mathematics Subject Classification: 54HJ5, 54C20, 54H20. Copyright© 1986 Rocky Mountain Mathematics Consortium 837 PROOF. By 2.5, we may consider {X, 1C) as a bG-space. By the results of [19], {X, 1C) can equivariantly be embedded in a compact Hausdorff bG-space (X, ft). Now consider X as a G-space and observe that, on X, the action of bG, hence the induced action of G, is equicontinuous. REMARK 2.14. If (X, 1C) is as in 2.13, then we may assume that X has the same weight as X: ... (X) = w(X). This follows immediately from [19; Proposition 2.10] because we consider bG-spaces, and bG has countable Lindel6f degree. A similar reasoning shows that, also, the maximal Gcompactifi.cation f3c (X, 1C) is equicontinuous. 3. £-admissible subsets. Again, we assume that, unless stated otherwise, G is an arbitrary topological group. 3.1. The following construction is standard in Topological Dynamics, see [9; 4.20]. Let (X, 1C) be a compact Hausdorff G-space, and let d/I denote the (unique) uniformity for X. With coordinate wise action, G also acts on X x X, and, since each a Ed/I is a subset of X x X, the expression Ga = {(tx, ty) : t E G and (x, y) ea} makes sense. Let Qx = n {Ga; a Ed/I}.
doi:10.1216/rmj-1986-16-4-837 fatcat:pvgshxht25gq5iscqh3j2h6r3m