This section of the book contains several results, all of which are about changing the topology on a G–space X. Given an action a : G × X → X by a Polish group G, we would like to put a new topology t on X such that: (X,t) is “nice”, e.g., metrizable or Polish; a is continuous with respect to t; and some given a–invariant set I ⊆ X is t–open (or closed or clopen). We have two different types of theorems, one type which says that there is a new topology which makes the action continuous, the other type which says that given a continuous action, there is a new, finer topology in which the action is still continuous and which makes an invariant set open. We have already seen an instance of such a result in 2.6.6, from which it follows that if G is Polish and X a Borel G–space, then we can put on X a separable metrizable topology having the same Borel structure in which the action is continuous.

In §5.1, 5.2 we consider the most important case, that of Borel–measurable actions and Borel invariant sets. In §5.3, 5.4 we consider more general actions and invariant sets.

While we will frequently be changing the topology on X, we will never change the topology on G. (In the light of 1.2.6 and the remarks following it, we could not change the topology on G even if we wanted to.)