# Tag Info

Accepted

### Is this group isomorphic to the real numbers?

The first and third properties suffice. The structure theorem implies that a connected locally compact abelian group $A$ is isomorphic to $\mathbb{R}^n \times K$ where $K$ is compact and connected. ...
Accepted

### Pontrjagin duality for a topological ring

Suppose $G$ is any locally compact abelian group. Then $\mathbb{Z}\times G$ is a locally compact ring with multiplication $(a,g)\cdot(b,h)=(ab,ah+bg)$ (in other words, take $1\in\mathbb{Z}$ as the ...
Accepted

### Circle with infinite corners

You are assuming that if you zoom in far enough on the circle, then you will be able to observe a "corner" where two lines meet to form a vertex. In other words, you are treating a line ...

### Topology induced by the completion of a topological group

$\def\scrB{\mathscr{B}} \def\scrP{\mathscr{P}} \def\scrN{\mathscr{N}} \def\c{\mathrm{c}}$I will give a more detailed explanation after GEdgar's answer. Most importantly, it is not immediately obvious ...
Accepted

### Consequences of differing definitions of topological groups

The second condition is equivalent to the condition that $G$ is Hausdorff, as others have said, and it does not follow from the first. It's true in all interesting examples and is a very mild ...
1 vote
Accepted

### Pontlyagin dual of direct sum,$\widehat{\bigoplus_{i\in \Lambda} M_i} = \prod_{i\in \Lambda} \hat{M_i}.$

I will assume the groups are discrete, as it simplifies some of the proof. Note that if you don't assume anything then you can not use the Pontryagin duality (you need at least that the groups are ...
1 vote
Accepted

1 vote
Accepted

1 vote
Accepted

### Openness of canonical projection of quotient topological group

This has nothing to do with topology. Given a group $G$ and a subgroup $H$ of $G$, one defines $G/H$ to be the set of left cosets of $H$ in $G$. A left coset of $H$ in $G$ is a subset of $G$ having ...
1 vote

### Is $\hat{G}$ is complete with respect to the induced topology of $G$?

(As I commented on Paul Sinclair's answer, their hint doesn't work to prove completeness of the completion.) I will assume first-countability. We imitate the proof of the completeness of the ...
1 vote

### Sum of Cauchy sequences in an abelian topological group (first countability hypothesis)

The reason why Atiyah and MacDonald restrict to first-countable topological abelian groups is because otherwise to construct the completion one would need to use the theory of filters (and other ...

Only top scored, non community-wiki answers of a minimum length are eligible