Questions tagged [topological-groups]
A topological group is a group endowed with a topology such that the group operation and inversion are continuous maps. They are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis.
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Different notion of almost periodicity using almost periods
I was recently trying to make sense of the definition of Besicovitch almost periodicity. Motivated by the many equivalent definitions of Bohr almost periodicity, I was wondering whether the same holds ...
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Running into strange conclusions when using the modular character to define a right Haar measure
In the following, $G$ is a locally compact, $\sigma$-compact metrizable group, and $m$ a left Haar measure. Exercise 10.5 in Einseidler & Ward's functional analysis book asks us to show that:
$$m^{...
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Continuity of the multiplication and inversion in the definition of topological group
A topological group $(G,\, *\,,\, e,\, {\cal{T}})$ is a topological space $(G,\, {\cal{T}})$ that is also a group whose unity element is $ e$ and both the multiplication
$$
\mu:\quad G \times G \...
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Proof that the underlying group of the free topological group is free
I am studying Arhangel'skii and Tkachenko's book on topological groups. In the chapter on free topological groups they offer a long and involved proof that the underlying group of the free topological ...
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Structure theorem for profinite abelian groups
If $G$ is a finitely generated profinite abelian group then
$$
G\cong \prod_p \mathbb Z_p^{a(p)}\times\prod_n(\mathbb Z/n\mathbb Z)^{b(n)}
$$
where $a(p), b(n)$ are integers and $a(p)=b(n)=0$ for all ...
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Defining the completion of a group can be done only using Cauchy sequences
Let $G$ be a group, in Atiyah & MacDonald's Commutative Algebra it says that
Assume for simplicity that $0\in G$ has a countable fundamental system
of neighborhoods. The completion $\hat G$ of $G$...
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Which commutative rings are endomorphism rings of some topological Abelian group?
I was thinking earlier today about how many commutative unital rings I could hit by considering the endomorphism rings of topological Abelian groups.
For example, the group $(\mathbb{R}, +, \tau^R)$ ...
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Every simple topological group is either discrete or connected [closed]
I read the claim in a preprint that "A simple topological group is either discrete or connected". However, the explanation given was "a connected component of a topological group that ...
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How to show isomorphism between an adele ring on a number field and its Pontryagin dual?
A book that I'm currently reading (Fourier Analysis on Number Fields by Dinakar Ramakrishnana, Robert J. Valenza) claims that a continuous homomorphism
$\mathbb{A}_k \to \widehat{\mathbb{A}_k}$ given ...
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Necessary condition for an isomorphism of modules to be a homeomorphism
Let $M$ and $N$ be two filtered $A$-modules with filtrations $(M_n)$ and $(N_n)$ and $u\colon M\to N$ be a morphism of modules. We can show that if $u(M_n)=N_n$ and $u$ is an isomorphism then $u$ is ...
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Every Cauchy sequence in $M/N$ converges.
Let $M$ be a filtered module with filtration $(M_k)$ and $N$ be a submodule of $M$. Then the filtration on $M/N$ is given by $(P_k)$ where $P_k=(M_k+N)/N$. We will show that if $M$ is complete and ...
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Generalization of the fact that for $f: S^1 \rightarrow \Bbb R$ continuous, $\exists x \in S^1 : f(x) = f(-x)$
While going through James Munkres' book on topology I came across the following problem:
Let $f: S^1 \rightarrow \Bbb R$ a continuous function. Show that there exists a point $x \in S^1$ st. $f(x)=f(-...
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Can a compact group have an infinite sequence of closed subgroups?
If $G$ is a copmact Hausdorff group then can there be, in some case of $G$, a sequence $G_i$, $i\ge0$ such that $G_0=G$ and $G_{n+1}$ is a closed proper subgroup of $G_n$ for all $n\ge0$ (proper ...
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Closed subsets in Fell topology on $\mathcal{P}(G)$
I am currently working with a discrete group $G$. I am considering a set $A \subseteq \mathcal{P}(G)$ of finite subsets $F \subseteq G$ whose cardinalities are uniformly bounded. I am having troubles ...
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Profinite completion of profinite groups
I was trying to prove that $\mathbb{R}/\mathbb{Z}$ cannot be a galoisgroup for any extension.
My plan for this was to show that $\mathbb{R}$ is a divisible group and that every quotient of a divisible ...
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