Topological and Borel compactifications of Polish G-spaces

We investigate the relationships between topological and Borel G-spaces, where G is a Polish group. We show that every Polish G-space can be topologically and equivariantly embedded into a compact Polish G-space iff G is locally compact. This answers a question of Kechris. It also provides a striking contrast to the recent result of Becker and Kechris which states that every Borel G-space can be Borel-embedded into a compact Polish G-space.