# 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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### Can the topology of a group be defined in terms of the group operation?

I am trying to digest the concept of a topological group. So far it makes sense to me that some structures (where the topology is already given), may have a continuous operation that also satisfies ...
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### Obtaining $\mathscr{B}G$ from the topological groupoid $BG$; which notion of "nerve" of a topological groupoid/category should be used?

For $G$ a group without topology (or a discrete topological group), let $BG$ denote the groupoid with one object and morphisms given by $G$. Then, as described at this nLab page, the geometric ...
1 vote
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### smooth differential forms on manifold with boundary

What is the definition of smooth differential forms on smooth manifold with boundary? Could someone please provide an example?
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### Is there a simple abstract reason why a profinite group is an inverse limit of finite groups?

Let $G$ be a profinite group (defined as a Hausdorff, compact, totally disconnected topological group). Suppose you know that as a profinite set, it's an inverse limit of finite sets. Is there an ...
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Call a subset of R2 radially open iff it contains an open line segment in each direction about each of its points. Show that the collection of radially open sets is a topology for R2 Compare this ...
1 vote
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### Is the universal $G$-bundle functorial in $G$?

We know that the classifying space construction $G \mapsto BG$ gives a functor from topological groups to spaces. I was wondering if the whole construction of the universal bundle is functorial in $G$?...
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### Group generated topologically

Let $G$ be a topological group and $X,Y$ subsets such that $N = \overline{\langle X \rangle}$ and $G/N = \overline{\langle \pi(Y) \rangle}$. Show that $G=\overline{\langle X,Y \rangle}$. In that ...
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### Is the space of distinct triples homeomorphic to a union of products?

$\newcommand{\S}{\mathbb{S}^1}$Let $M=\{(x,y,z) \in (\S)^3 \, |\,\, x,y,z \,\,\text{are distinct}\}$. Is $M$ homeomorphic to a finite union of products of one-dimensional manifolds? I think $M$ is ...
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### Is the space $\mathbb{PR}^3$ homeomorphic to $\mathbb{PR}^2\times S^1$?

In this Wiki article it is described how the $SO(3)$ is homeomorphic to the projective space $\mathbb{RP}^3$. I would suggest another way which I hope it works. On $S^2$ one may take any direction (...
1 vote
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### the kernel of the covering map is isomorphic to the covering transformation group of topological group

I'm trying to prove the following statement: Suppose $G$ and $\tilde G$ is connected and locally path connected topological groups, and $p:\tilde G\rightarrow G$ is a covering map and a homomorphism ...
1 vote
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### Prove that wedge sum of $S^2$ and $S^1$ homotopy equivalent to the union of $S^2$ and it's diameter

We have 2 topological spaces: wedge sum of $S^2$ and $S^1$ and the union of $S^2$ and it's diameter. Should prove that their spaces homotopy equivalent. My ideas: we can see that these 2 spaces are ...
1 vote
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### $G$ be loacally compact abelian group with $G,\hat{G}$ both sigma compact. Let $f\in L^2(G)$ and $\psi_n(x)=\int_{C_n}\hat{f}(\chi)\chi(x)\ d\chi$.

Let $G$ be a locally compact abelian, $T_2$ group such that $G,\hat{G}$ both are $\sigma$-compact with $G=\bigcup K_n$ and $\hat{G}=\bigcup C_n$ where $K_n,C_n$ are increasing compact subsets of $G$ ...
1 vote
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### Equivalent topologies on the Tate Module

This is a question from Chapter III, section 7 of Silverman's Arithmetic of Elliptic Curves. The section introduces the Tate module and then the author makes the offhand remark: Since each $E[\ell^n]$...
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### Is the Haar measure of the boundary of an open subset of the $k$-torus zero?

I have just recently started reading the basics of topological groups and Haar measures, and have become really curious about when the boundary of a non-empty open subset $U$ of a compact abelian ...
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### Cylinder as a quotient space

Consider in $\mathbb{R}^2$ the map $g(x,y) = (x+1,-y)$. It is easy to see that $G = \langle g\rangle$ acts proper and discontinously on $\mathbb{R}^2$, so $\pi : \mathbb{R}^2 \to \mathbb{R}^2/G$ is a ...
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### Pointwise convergence V.S. Compactly convergence on Dual group

Let $G$ be a LCA group (Locally compact and Abelian). $\hat{G}$ denote the dual group of $G$. In most of all text books, the topology of $\hat{G}$ is given by the open-compact topology, namely, a net ...
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### Existence of a certain type of function on locally compact groups

I have seen it stated that on a locally compact group $G$ with $\mu$ its (left) Haar measure, there exists a positive, compactly supported function $f\in C_c(G)$ with $\int_G f(s)d\mu(s)=1$ satisfying ...
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### Group/H-space structures for standard models of classifying spaces?

Let $G$ be a commutative topological group. May in his textbook, A concise course in algebraic topology, gives a model of the classifying space $BG$ so that it is a commutative topological group ...
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### A subgroup of full measure is dense given a haar measure

I want to know why if $\mu$ is a haar measure on a compact $G$ and $\mu(A)=\mu(G)$ then $A$ is dense in $G$. This fact is mentioned in the wikipedia page, but I couldn't find a proof for it.
1 vote
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### Is a certain map $\mathbb{R}\to \widehat{\mathbb{R}}$ a homeomorphism?

I'm studying harmonic analysis, and I'm trying to understand the fact that $$\phi: \mathbb{R}\to \widehat{\mathbb{R}}: s \mapsto (t \mapsto \exp(2\pi i st))$$ is an isomorphism of topological groups (...
1 vote
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### Are admissible topological rings compactly generated?

Suppose $R$ is a (commutative unital) topological ring which is admissible in the sense of Stacks 07E8: it is complete, Hausdorff, admits a fundamental system of neighborhoods of zero consisting of ...
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### Find connected closed subgroup of $(\mathbb C,+)$

I am asked to show that the only connected closed subgroup of $(\mathbb C,+)$ are $\{0\}$, $\mathbb C$, or a line passing through the origin. Since the subgroup is connected, $0$ is a limit point. It ...
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### If $U$ gives fundamental system of $0$ of $G$, and $U$ also gives fundamental system of $0$ of $G'$, then, does $G$ and $G'$ has the same topology?

Let $G$ and $G'$ be two topological abelian groups with the same underlying abelian group. If $U$ gives fundamental system of $0$ of $G$, and $U$ also gives fundamental system of $0$ of $G'$, then, ...
66 views

### Necessary and sufficient condition for metrizability of topological group , module , ring

We know that a topological vector space is metrizable iff it has a countable local base and in general a topological space is metrizable iff it is $T_3$ and has a countably locally finite basis. Now ...
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### A measure theoretic problem related to induced representations

I met such a rather concrete measure theoretic problem while dealing with induced representations of $p$-adic groups. So let $G$ be a unimodular locally compact group, with $P$ its closed subgroup (...
1 vote
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### Let $G$ be locally compact abelian, $T_2$ and $f\in L^1(G)$. Define $\mu(A)=\int_A f(x)\ dx$. Prove $\lVert \mu\rVert=\lVert f\rVert_1$

Here $\mu$ becomes a complex measure and $\lVert \mu\rVert =|\mu|(G)$ is the total variation norm of $\mu$. We have to show $|\mu|(G)=\int\limits_G |f(x)|\ dx$ Let $\{A_n\}$ be a partition of $G$. ...
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### The centralizer of a set $M$ is a closed subgroup

This is a exercice from the San Martin's book "Lie Groups". Let $G$ be a Hausdorff topological group. Show that the centralizer $$\{g \in G : \forall x \in M,gx=xg\}$$ of the set $M$ is a ...
1 vote
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### Closed subgroups of profinite groups and basis of neighbourhoods

Let $G$ be a profinite group with a basis of neighbourhoods $U_n$ of normal subgroups. Furthermore let $H\subset G$ be a closed subgroup. Then we can define the open subgroup $H_n:= H\cdot U_n$. Is ...
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1 vote
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### Injectivity of divisible locally compact abelian groups

Are divisible locally compact abelian groups injective as objects of the quasi-abelian category of locally compact abelian groups ? At the very least, if $D$ is a divisible locally compact abelian ...
1 vote
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### Is any finite index subgroup of multiplicative group of p-adic field open?

I found that any finite index subgroup of multiplicative group of p-adic integer is open. But i don't know how to prove that any finite index subgroup of multiplicative group of p-adic field is open ...
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### What are the epimorphisms in the category of topological groups?

A morphism $f: X \to Y$ is an epimorphism if for all $g, h: Y \to Z$, if $g \circ f = h \circ f$ then $g = h$. The epimorphisms in the category of groups are the surjective group homomorphisms. The ...
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### Topology by a collection of cosets which makes product continuous

Let $G$ be any group. Let $\mathcal{F}$ be a family of some subgroups of $G$, along with $G$, which is closed under finite intersection. Then all left cosets $\{xH : x \in G, H\in \mathcal{F} \}$ ...
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### Groups with "same" finite-index normal subgroups

If $G, G'$ are two groups whose categories of finite-indexed normal subgroups are equivalent. Then, are they or their profinite completions isomorphic ? If instead G, G' are topological groups (...
1 vote
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### Is there a compact metrizable non-abelian torsion-free group?

It is quite difficult to google for non-abelian torsion-free groups. I am primarily interested in metrizable (''not too big'') compact groups. However, even if we drop the metrizability condition I do ...