Polish Groups Topological Groups a Topological Group Is a Group G with a Topology on G for Which the Operations

These notes introduce the reader to Polish groups|topological groups in which the underlying space is Polish. No background in topological groups is assumed, but I will present the material fairly rapidly and leave quite a lot for the reader to check on his or her own. My main motivation for writing such notes arises from the applications and connections in model theory, where closed subgroups G 6 Sym are studied, the topology arising (as in the Galois theory of innnite eld extensions) from consideration of pointwise stabilizers of nite sets. Such groups are Polish with the small subgroup property and these are the main object of study. However, much of the theory carries through to Polish groups in general. If there is any original material at all in these notes it probably concerns the generalization of the small index property|familiar nowadays to model theorists in the case of the small subgroup property|to Polish groups in general. are continuous. (Use the product topology for G 2 .) Thus the notion`topological group' arises by just combining the deenition of a group with the topological notion of continuity in the most obvious way. It is quite remarkable how these two seemingly quite diierent ideas interact with consequences both for the topology on G and for the group theoretic structure of G. To start with, by composing the continuous maps h 7 ! (h; g) and (h; g) 7 ! hg, it is clear that in such a group that the right-translation map h 7 ! hg is a continuous, and (by considering g ?1) a homeomorphism G ! G. In particular, if H < G is an open subgroup then every coset Hg of H is open. In the same way, the map h 7 ! h g = g ?1 hg is also a homeomorphism G ! G. So conjugates of open subgroups are open.