1

I am currently reading about completions of a topological group from Atiyah, MacDonald- Commutative Algebra.

Some prerequisites :- A Cauchy sequence in a topological group $G$ is defined to be a sequence $({x_n})_{n\in \mathbb{N}}$ of elements of $G$ such that, for any neighborhood $U$ of $0$, there exists an integer $s(U)$ with the property that $x_m - x_n \in U $for all $m,n \geq s(U)$. Two Cauchy sequences are equivalent if $x_m - y_m \to 0$ in $G$. The set of all equivalence classes of Cauchy sequences is denoted by $\hat{G}$.

The set $\hat{G}$ is naturally a group. The book mentions that there exists a group homomorphism $$\phi:G\to\hat{G} $$ that sends $x\in G$ to the equivalence class of the constant Cauchy sequence $(x)_{n\in\mathbb{N}}$. But it also says that this map is continuous without telling what the topology on $\hat{G}$ is.

Can anyone help me understand the topology on $\hat{G}$?

KReiser
  • 65,137
Neo
  • 11
  • 2
  • see https://math.stackexchange.com/questions/192808/topology-induced-by-the-completion-of-a-topological-group?rq=1 – Stefan May 30 '20 at 17:07
  • @Stefan The user has not provided a reference source for his argument. Do you know of any such source? – Neo May 30 '20 at 18:35
  • 1
    This topic is covered in Bourbaki, General Topology (§3.4), but probably not in a way that is immediately accessible to you. – Stefan May 30 '20 at 22:21
  • My books all give different constructions for the completion. Of course in general it's complicated, as we need to consider Cauchy filters (not just Cauchy sequences) and a group can have two natural uniformities if it's not Abelian (maybe in this book all groups are Abelian, removing that complication?). Engelking has a chapter on uniform spaces, and there are quite a few books that will cover this too. – Henno Brandsma May 30 '20 at 22:22

0 Answers0