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 ...
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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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Radilly open topology [closed]

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 ...
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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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Unitary dualization functor continuous?

Let $G$ be a topological group and denote its unitary dual by $\hat{G}:=\{\pi:G\to\text{U}(\mathcal{H})\text{ irreducible unitary representation}\}/_\cong$. If $H$ is another topological group and $\...
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Pontryagin dual and unitary dual of abelian group homeomorphic?

Let $G$ be an abelian topological group, $G^\vee:=\{f\in\text{Hom}(G,\mathbb{T})\text{ continuous}\}$ its Pontryagin dual and $\widehat{G}:=\{\pi:G\to\text{U}(\mathcal{H})\text{ irreducible unitary ...
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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 (...
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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 ...
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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 ...
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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 $G$ ...
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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.
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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 (...
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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, ...
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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 (...
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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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2 votes
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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 ...
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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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What does it mean that a $G$-space embeds into its homotopy quotient as the fiber over the basepoint of the classifying space of $G$?

Let $G$ be a topological group and $M$ be a $G$-space. Let $EG \rightarrow BG$ be a universal $G$-bundle and let $M_G$ be the homotopy quotient $(EG \times M)/G$. What does it mean that $M$ embeds in $...
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3 answers
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To show [0,1] is not homogeneous

How can we show that [0,1] is not homogeneous topological space? So far what I have thought is as follows:- If it is homogenous then there exists a homeomorphism between every pair of points in it. I ...
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Can we develop differential theory out of Euclidean spaces, just with the structure of topological groups?

It is well known to all that we define the differential of the map $f:R^n\to R^m$ at $x$ to be the linear map $Df(x)$, which satisfies that $$ f(x+h)-f(x)= Df(x)h+o(h) $$ Now I think linear maps as ...
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What are the closed subgroups of $SO(3)$?

What are the closed subgroups of $SO(3)$? I think that it is a textbook question but I haven't studied it anywhere. I haven't studied Lie theory. I am thinking about the question from a basic abstract ...
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When is the trivial subgroup, $1$, isolated in the set of all closed subgroups of $G$?

Let $G$ be a compact group. When is the trivial subgroup, $1$, isolated in the set of all closed subgroups of $G$? I think that the isolated identity is called "No small subgroup". Either ...
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Changing variable in topological groups

Let $G \subset GL_n(F)$ be a locally compact group endowed with its Haar measure. Two typical automorphism of $G$, involutions even, are the transposition and inversion. Is it clear that we have the ...
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Is Homeo(M) locally path-connected for a general topological manifold?

I am wondering if there exists a closed topological manifold for which Homeo$(M)$ is not locally path-connected. If $M$ admits a smooth structure, then one can prove that Homeo$(M)$ is in fact locally ...
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Is $\mathcal{O}=\{\pi_{v,w}| \ (\pi,H)\in Rep_f^s(G),\ v,w\in H\}$ dense in $C(G)$? where $G$ is compact, $T_2$ group

Let $Rep_f^s(G)$ be the set of all finite dimensional, strongly continuous hilbert space representations of compact, $T_2$ group $G$. For $(\pi,H)\in Rep_f^s(G)$ and $v,w\in H$ define $\pi_{v,w}(g):=\...
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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 ...
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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 (...
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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 ...
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Convolution on a locally compact group is associative

Consider the following fragment from Folland's book "A course in abstract harmonic analysis" (question is below the image). Can someone explain why the boxed equality is true? Don't we need ...
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Does $(x,y)\mapsto \phi_x(y)$ vanish outside some compact set in $G\times H$?

Suppose that $G$ and $H$ are locally compact Hausdorff groups. EDIT: As Eric Wofsey pointed out in the comments; the group structures can probably be ignored for this question. Assume that for each $x\...
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Proof that the Dual of an LCA Topological Group is LC with the compact-open topology

Let G be a LCA Topological Group, I define it's dual group, $\Gamma$, by the group of continuous characters $$\gamma:G\to\mathbb{T}$$ where $\mathbb{T}$ is the complex unit circle. I want to equip ...
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1 vote
1 answer
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Relation between fundamental system of neighborhood and basis of topological space

What is the difference and relation between fundamental system of neighborhood and basis of topological space ? Definition. A base is a collection $B$ of sets such that every set in the topology can ...
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Krull topology is discrete if only if the exntension is finite

Let $M/K$ be arbitrary extension. Then, we can define Krull topology on $Gal(M/K)$ by taking {$Gal(M/F)|K⊆F⊆M,[F:K]<∞$} as fundamental system of neighborhood of $0$. I guess $Gal(M/K)$ has discrete ...
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$p^n \Bbb{Z}_p$ is closed subgroup of $ \Bbb{Z}_p$

I want to check $p^n \Bbb{Z}_p$ is closed subgroup of $ \Bbb{Z}_p$. It is clear that $p^n \Bbb{Z}_p$ is subgroup of $ \Bbb{Z}_p$, I want to prove $p^n \Bbb{Z}_p$ is closed in $\Bbb{Z}_p$. So, I only ...
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