Notes on Filtrations, Topologies, and Completions

(i) G is Hausdorff. (ii) Points in G are closed. (iii) The Gi intersect in {e}. Of course, (i) implies (ii) by general topology. For (ii) implies (i), the diagonal in G×G is μ(e), where μ(g, h) = gh. Since G−Gi is the union of the cosets gGi with g / ∈ Gi, G−Gi is open, hence G−Gi is both open and closed, hence so is Gi. Now (iii) clearly implies (ii). If x is in all Gi and is not e, then there are no open neighborhoods separating e and x, so G is not Hausdorff. 3. If H ⊃ Gi, then H − Gi is open since it is the union of the cosets hGi, h ∈ H −Gi and, similarly, H is also closed. 4. G/ ∩Gi is the associated Hausdorff group of G. 5. Consider the canonical map γ : G −→ limG/Gi, obtained from the quotient homomorphisms γi : G −→ G/Gi. Give the target the inverse limit topology, where the G/Gi are discrete. Then γ is continuous since γ −1 i (eGi) = Gi. If γ is a bijection, then it is a homeomorphism. Indeed, the Gi then give a fundamental system of neighborhoods of the identity in both. We say that G is complete when this holds. 6. Define the completion of G to be Ĝ = limG/Gi; it is more accurate to view the map γ : G −→ Ĝ as the completion of G. Let Ĝi be the kernel of Ĝ −→ G/Gi. This gives Ĝ a decreasing filtration, and the topology on Ĝ is the same as the topology associated to this filtration. Moreover, G/Gn ∼= Ĝ/Ĝn.