# Tagged Questions

A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).

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### If G is a sequential topological group, must G x G be sequential?

Using standard definitions, the topological space $Y$ is sequential if for each nonclosed $A \subset Y$, there exists a convergent sequence $a_{1}$ , $a_{2}$,...$\rightarrow b$ so that $a_{n} \in A$ ...
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### Layman question: A dense subgroup with completion not isomorphic to the big (pro-p) group?

This is unlikely a research level question... one that would be answered in a blink of an eye, rather...it is an (early) exercise from the book "Analytic Pro-p groups". But since no reply was received ...
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### Naive question on adelic groups

The ever-reliable Wikipedia says: ... an adelic algebraic group is a semitopological group defined by... No more details are given, and I was wondering if the multiplication only being ...
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### Contractible topological groups

Does there exist a Hausdorff topological group which is contractible and of finite covering dimension but which is not homeomorphic to $\mathbb{R}^n$ for some $n$?
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### Strongly Complete Profinite Groups.

I've been reading about profinite groups and have encountered the notion of strong completeness. I.e. that a profinite group $G$ is strongly complete if it is isomorphic to it's profinite completion ...
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### Mean value theorems for the Haar integral?

Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral? In general, are there mean value theorems ...
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### A dynamical property of automorphisms of a locally compact group

Let $G$ be a Hausdorff locally compact group and let $\alpha$ be an automorphism of $G$. Say $\alpha$ is (forwards) topologically recurrent if for all $g \in G$ and all neighbourhoods $O$ of $g$, the ...
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### When is a valued field second-countable?

Let $R$ be a valuation ring, with fraction field $K$ and residue field $k$. Denote by $\Gamma=K^{\times}/R^{\times}$ the valuation group (assumed nontrivial). The valuation $v:K^{\times}\to\Gamma$ ...
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### Why groups that admit Folner Sequences are amenable

I've been looking at Folner's Condition recently, and I'm struggling to find a proof for why the existence of a Folner sequence on a locally compact group implies that it is amenable (and the converse ...
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Let $G$ be a locally compact Hausdorff group. It is known that $G$ can be topologically embedded in $W^{\ast}(G)$ , its universal $W^{\ast}$-algebra (with the $\sigma$-weak topology). An element $T \... 1answer 381 views ### Are the categories of representations of G and C*(G) isomorphic? Let G be a locally compact Hausdorff group, and C*(G) the full group C* algebra. I found in some books that "representation theory of both is the same". Can this be expressed as "the categories are ... 0answers 271 views ### 'Infinitesimal' elements of a topological group Let$G$be a topological group, and let$M$be the intersection of all conjugacy-invariant neighbourhoods of the identity in$G$(in other words, the set of elements that can be taken arbitarily close ... 1answer 821 views ### Proof that the Pontryagin dual of a topological group is a topological group I'm looking for a proof that the Pontryagin dual$G^*$of a topological group$G$is a topological group. It's very easy to prove that$G^*$is a group, my troubles are in proving that the map$G^* \...
I am interested in examples of the following property, where $G$ is a non-discrete locally compact topological group: (*) The open normal subgroups of $G$ have trivial intersection, but $G$ has an ...
Let $\mathbf{R} = (R,\mathcal{T},+,\cdot,0,1)$ be a compact Hausdorff topological unitary ring, and consider the set $I(\mathbf{R}) := \{ e \in R \mid e \cdot e = e \}$ (of idempotents in $\mathbf{R}$)...