17
votes
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. ...
- 409k
2
votes
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 ...
- 323k
2
votes
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 ...
- 4,662
2
votes
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 ...
2
votes
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 ...
- 409k
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 ...
- 13.3k
1
vote
Accepted
Doubts in the proof that a subgroup $H$ of a Lie group $G$ that is a submanifold of $G$ is (topologically) closed in $G$
$(1)$ is true in the following sense:
Recall that an open set in $U$ is defined to be $V \cap U$ where $V$ is open in $G$. Then because $U$ is open in $G$, $V \cap U$ is also open in $G$. Suppose $g \...
- 1,895
1
vote
Accepted
Embedding of topological group in its group of self-homeomorphisms
It is correct. You can even define
$$\beta : \text{Map}(G,G) \to G, \beta(f) = f(1) .$$
Then $\beta^{-1}(U) = M(\{1\},U)$ which is open in $\text{Map}(G,G)$. Clearly $\beta \mid_{L_G}$ is continuous ...
- 71.5k
1
vote
Accepted
Quotient by compact topological group has closed projection
Let $\mu \colon G \times X \to X$ the action map, this map is closed because $G$ is compact.
Let $C$ be a closed subset of $X$, then $G \times C$ is a closed subset of $G \times X$ so is $\mu(G \times ...
- 225
1
vote
Accepted
Universal property of the completion of a first-countable abelian topological group
Let $W\subset H$ be an open neighborhood of $0$ such that $W+W\subset V,$ and let $U:=\psi^{-1}(W).$
For all $n,m\gg0,$ $x_{nm}\in U,$ i.e. $\psi(x_{nm})\in W.$
Therefore, for all $n\gg0,$ $\lim_m\...
- 27k
1
vote
Accepted
Homogeneous space with finite invariant measure
Yes, if $H$ is compact, then it admits a finite Haar measure $\mu_H$. Let $\mu_{G/H}$ denote the finite measure on $G/H$, we can define a measure on $G$ by
$$\int f d\mu_G = \int_{G/H} \mathbb{E}_H(f) ...
- 13.3k
1
vote
Accepted
Sum of the completion of a topological abelian group is continuous
[*Update, corrected a problem with the original argument.]
Unfortunately point #3 in the question appears incorrect, since certainly for $s_1,s_2\in S$ and $x\in A$ we cannot conclude $s_1+(s_2+x)\in ...
- 3,757
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 ...
- 71.5k
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 ...
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