A Note on the Effros Theorem

(g, x) → gx : G × X → X such that ex = x for every x in X and g(hx) = (gh)x for g and h in G and x in X (here e denotes the neutral element of G). It is easily seen that for each g in G the map x → gx is a homeomorphism of X whose inverse is the map x → g−1x . If x belongs to X and U is a subset of G, then U x = {gx : g ∈ U}. The action of G on X is transitive if Gx = X for every x in X . It is micro-transitive if for every x in X and every neighborhood U of e in G the set U x is a neighborhood of x in X . A metric on a space X is admissible if it generates the topology on X . A space is Polish if it has an admissible complete metric.