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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Research Sources for $SL(2,R)$

Can anyone guide me to a good site for the special linear group $SL(2,R)$, especially one that goes deep into its subgroup and normal subgroup? Book recommendations would be great too.
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If the action of a group $G$ on $\mathbb{R}$ is properly discontinuous then G is isomorph to $\mathbb{Z}$?

Let $G$ be a topological group, acts on a topological space $X$, such that the map $f: G \times X \rightarrow X:(g,x)\mapsto g*x$ is continuous. We say that this action is $properly\;discontinuous$ ...
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G is a topological group acts on topological space $X$, is $f_{g}:X\rightarrow X, x\rightarrow g*x$ continuous?

Let $G$ be a topological group acts on the topological space $X$, for an elememt $g\in G$, let's define the map $f:X\rightarrow X, f(x)=g*x$. I am trying to find if $f$ is continuous? my best ...
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67 views

$G$ finite group acts freely on top. sp. $X$, can we find for every $x\in X$ an open neighborhood such that:

Let $G$ be finite topological group, and acts freely over the hausdorff topological space $X$, i want to prove that every element $x$ in $X$ has an open neighborhood $U_x$ such that: $g\star ...
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Why isn't there interest in nontrivial, nondiscrete topologies on finite groups?

A topology on a group is required to be compatible with the group structure (multiplication must be a continuous map $G\times G\to G$ and inversion must be continuous). I've only ever seen the ...
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102 views

Topological structure of a quotient of ${\rm{SU}}(2)\times{\rm{SU}}(2)$

I'm trying to understand the topology of the product of two three dimensional spheres $\mathbb{S}^3\times \mathbb{S}^3$ quotiented by the action of $\pm 1$ sending a pair of points $(x,y)$ to the ...
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Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group?

And what else can be said, if so? In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (It also has a two-sided ...
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371 views

Basics of Haar measure

Suppose $G$ is a locally compact group. Then $G$ has a left-invariant measure $dg$, say, which means that $$\int f (hg) dg = \int f(g) fg$$ for any test function integrable on $G$. The ...
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open subsets in topological groups

I'm starting to study topological groups, and I noticed that Every single theorem in topological groups I have to use the following statement: Let $G$ be a topological group and U an open subset of ...
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137 views

Topological groups - Actions and Hausdorffness

I guess this problem is widely-known, but I couldn't finish it. If $X$ is a topological group (and compact), and $G$ a closed subgroup acting on $X$ by left translation, show that $X/G$ is Hausdorff. ...
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The real line with its additive group is a topological group?

Maybe it's a stupid question, I'm starting to study topological groups, I'm struggling to prove that the real line is a topological group with its additive group structure and Euclidean topology, ...
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149 views

Why are locally compact groups Weil complete?

Why are locally compact groups Weil complete? Note: A topological group $G$ is Weil complete if every left Cauchy net in $G$ is convergent. Thank you, and sorry if I have bad writing.
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2answers
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Are inversion and multiplicaton open?

If $G$ is a topological group, are inversion $G \to G$ and multiplication $G\times G \to G$ open mappings? More concretely, I try to show that division of complex numbers $$\{(z,w) \in \mathbb{C}^2;\; ...
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1answer
47 views

Why is the topology of characters determined by the open sets containing the trivial character?

Let $G$ be an abelian topological group, and let $\hat G$ denote the set of characters on $G$. Why is it true that if one has a topological basis of for the trivial character (say the topological of ...
7
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139 views

Generators of $GL_n(\Bbb Z)$ and $GL_n(\Bbb Z_p)$

Let $\mathbb{Z}_p$ denote the $p$-adic integers. I know that the groups $GL_n(\mathbb{Z})$ and $GL_n(\mathbb{Z}_p)$ are (topologically for the latter) finitely generated. My question is: what are the ...
3
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1answer
95 views

Induced topology on the homomorphic image of a topological group

I would like to do a small sanity check on the following situation: Let $\pi: G \rightarrow G'$ be a surjective homomorphism of topological groups. Let the topology of $G$ be given by a sequence of ...
4
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1answer
134 views

Consequences of Pontryagin Duality?

What are some interesting corollaries and consequences of the Pontryagin Duality theorem? My question can be taken as broadly as you'd like, even up to including any philosophy introduced specifically ...
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401 views

Local Isomorphism on Topological Groups

I'm currently studying Lie Groups by "Theory of Lie Groups I", C. Chevalley. He talks about Topological Groups on chapter two. To be more precise, on page 38 he presents two examples in order to show ...
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249 views

Is a quotient of a complete group always complete?

Let $\: \langle G,\cdot,\mathcal{T}\hspace{0.01 in} \rangle \:$ be a $\big($$\text{T}_0$$\big)$ topological group. $\;\;$ Let $H$ be a closed normal subgroup of $G$. Set $\;\; \mathbf{G} \: = \: ...
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430 views

Topology induced by the completion of a topological group

Let $G$ be an abelian topological group and let $\hat{G}$ be its completion, i.e. the group containing the equivalence classes of all Cauchy sequences of $G$. What exactly is the topology of ...
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Sum of Cauchy sequences is Cauchy in an Abelian Topological Group

Let $G$ be a topological abelian group and suppose $0$ has a countable fundamental system of neighborhoods. Let $(x_n),(y_n)$ be Cauchy sequences of $G$. Why is it true that $(x_n+y_n)$ is a Cauchy ...
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1answer
149 views

Two Lie groups which are isomorphic but not homeomorphic

I am looking for an example of two Lie groups which are isomorphic as groups but not homeomorphic as topological spaces. Or, even more interestingly, a proof that two such groups cannot exist. Does ...
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1answer
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Why is this quotient space not Hausdorff?

I am trying to show that the following space is not Hausdorff. Consider the topological space $S^1$, and let $r$ be an irrational number. Consider the action of $\mathbb{Z}$ on $S^1$ given by $$ ...
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Is every group a Galois group?

It is well-known that any finite group is the Galois group of a Galois extension. This follows from Cayley's theorem (as can be seen in this answer). This (linked) answer led me to the following ...
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Lie Groups: Identity Component

Main Problem Given a Lie group. The connected component of the identity is a Lie subgroup: It is a subgroup. It is open. How to check this using topological tools? Extra Problem The quotient ...
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Visualizing quotient groups: $\mathbb{R/Q}$

I was wondering about this. I know it is possible to visualize the quotient group $\mathbb{R}/\mathbb{Z}$ as a circle, and if you consider these as "topological groups", then this group (not ...
6
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1answer
127 views

Unipotent action of pro-$p$-group

Say $p$ and $\ell$ are distinct prime numbers. Let $G$ be a pro-$p$-group which acts continuously on a finite-dimensional $\mathbb{Q}_\ell$-vector space $V$. Assume that the action of $G$ on $V$ is ...
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1answer
260 views

Fourier transform of a measure

I'm a bit confused - How is the Fourier transform of a measure on a compact abelian group defined? specifically the Fourier transform of a measure on $\mathbb{T}$ the unit circle in the complex plain. ...
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1answer
107 views

Strongly complete profinite group

Let $G$ be a profinite group (or equivalently a compact and totally disconnected topological group ) with the property that all of its normal subgroups of finite index are open sets. Does this ...
6
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1answer
193 views

Addition and multiplication are continuous in the $I$-adic topology

Can you tell me if this is correct? Let $R$ be a ring and let it have the $I$-adic topology for some ideal $I$ in $R$. I want to show that $+: R \times R \to R$ is continuous at $(x_0, y_0)$. Proof: ...
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Adic Completion of a direct sum

This is a question related to this. Let $G= \mathbb Z / p \mathbb Z$ for some prime $p$. Let $A = \bigoplus_{n\in \mathbb N} G$, that is, all sequences in $G$ with all but finitely many terms zero. ...
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Quick question about Hausdorffness of completion

Let $G$ be the usual: a topological Abelian group with a topology induced by a countable neighbourhood basis $G_n$ of zero such that $G = G_1 \supset G_2 \supset \dots$. Let $\widehat{G}$ denote the ...
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1answer
108 views

Quick question about completions and inverse limits

Let $G$ be the usual: a topological Abelian group with a topology induced by a countable neighbourhood basis $G_n$ of zero such that $G = G_1 \supset G_2 \supset \dots$. Let $\widehat{G}$ denote the ...
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3answers
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Isomorphism from $\widehat{G}$ to $\displaystyle \lim_{\longleftarrow} G/G_n$

Let $G$ be a topological abelian group and let $\widehat{G}$ denote its completion (i.e. equivalence classes of Cauchy sequences). Let $G_n$ be a descending sequence of subgroups, i.e. $G = G_0 ...
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1answer
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Completion $\widehat{G}$ is a topological group

I want to verify that if $G$ is an abelian topological group then so is its completion $\widehat{G}$. Note that $G$ is not necessarily a metric space. Hence we define a sequence $x_n$ to be Cauchy if ...
7
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1answer
479 views

Intersection of all neighborhoods of zero is a subgroup

Let $G$ be a topological abelian group. Let $H$ be the intersection of all neighborhoods of zero. How is $H = \mathrm{cl}(\{0\})$? Isn't the closure of a set $A$ the smallest closed set containing ...
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1answer
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Continuous Actions and Homomorphisms

I am learning about the compact-open topology and have a small proposition I am struggling to prove. Let $G$ be a topological group, $X$ a compact, Hausdorff space, and $H(X)$, the homeomorphisms of ...
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Endomorphisms preserve Haar measure

I am having trouble following the argument in page 21 of P. Walters, Intro. to ergodic theory, of the following statement: Any continuous endomorphism on a compact group preserves Haar measure. ...
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Fourier series on $\mathbb T$ and $S^1$

From my lecture notes: "The notation $\mathbb T$ will be used for the additive circle and $S^1$ for the multiplicative circle." What I understand: As a topological group, $S^1$ has the subspace ...
7
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1answer
457 views

Homeomorphisms of X form a topological group

So I'm just learning about the compact-open topology and am trying to show that for a compact, Hausdorff space ,$X$, the group of homeomorphisms of $X$, $H(X)$, is a topological group with the compact ...
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1answer
123 views

a question on normal subgroup of $GL_n(\mathbb{C})$ and $GL_n(\mathbb{R})$

I am really sorry that I am not able to solve this one, thank you for your help.
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Can continuity of inverse be omitted from the definition of topological group?

According to Wikipedia, a topological group $G$ is a group and a topological space such that $$ (x,y) \mapsto xy$$ and $$ x \mapsto x^{-1}$$ are continuous. The second requirement follows from the ...
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175 views

topological group

Recently I'm interested in this open question: Must every star compact topological group be countably compact? star compactness ( which implies pseudocompactness ) = for every open cover $U$ of ...
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0answers
86 views

$1$-parameter subgroups in $GL_n(\mathbb{C})$

I came across this link on planetmath and a few facts on that link are confusing me. According to planetmath, any $1$-parameter subgroup in $GL_n(\mathbb{C})$ arises from the exponential map. That ...
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Definition of a topological module

A topological universal algebra of type $\Omega$ is a universal algebra $A$ of type $\Omega$ that is also a topological space, such that for any $n\!\in\!\mathbb{N}$ and any operation ...
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208 views

subgroup of connected locally compact group

I need a reference or a short proof for the following property: A nontrivial connected locally compact group $G$ contains an infinite abelian subgroup.
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245 views

Properties of Topological Groups

I'm working though William Basener's Topology and Its Applications and I have come across a problem I can't solve. The book defines a topological group as a group equipped with a topology where for ...
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Group structure on $\mathbb R P^n$

For which positive integers $n$ can $\mathbb R P^n$ be given the structure of a topological group? I believe that $\mathbb R P^n$ cannot be given a Lie group structure for even $n$, since then it is ...
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280 views

Example of profinite groups

Could someone help me with an simple example of a profinite group that is not the p-adics integers or a finite group? It's my first course on groups and the examples that I've found of profinite ...
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How can we find and categorize the subgroups of R?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}\newcommand{\Z}{\Bbb Z}$ What are all the subgroups of R = $(\R, +)$ and how can we categorize them? I started thinking about this question last night ...