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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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 $$ S^1\...
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Topological group: Multiplying two loops is homotopic to linking these paths?

Let G be a topological group and let $s_1$ and $s_2$ be loops in G (both loops are based at the identity e of G). Is it true that the loop $s_1s_2$ (where the multiplication is the one of the group ...
8
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2answers
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About connected Lie Groups

How can I prove that a connected Lie Group is generated by any neighborhood of the identity? The result is almost trivial for $R^n$ but I tried using the open subgroup generated by this neighborhood....
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2answers
986 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
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6answers
925 views

How to show path-connectedness of $GL(n,\mathbb{C})$

Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...
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1answer
731 views

Sum of Cauchy Sequences Cauchy?

Let $(X,+)$ be an abelian group and $d$ a metric on $X$. Suppose $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences. What conditions on the relation between the group operation and the metric are sufficient ...
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802 views

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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1answer
949 views

The group of invertible linear operators on a Banach space

Let $X$ be a Banach space. Let $G$ be the group of invertible linear operators from $X$ to itself. Now my questions are: If $G$ is equipped with the operator norm topology, how do you show that it ...
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Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

Could you please explain me the reason why they are isomorphic? Thanks, bye!
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961 views

$G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian

Hypothesis: Let $G$ be a topological group with identity element $e$. Let $\mu$ denote the multiplication mapping in $G$. Goal: Show that $\pi_1(G,e) = \pi(G)$ is an abelian group via the hint below....
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0answers
116 views

Explicit construction of Haar measure on a profinite group

Let $G$ be a profinite group. It is known that in $G$, every neighborhood of the identity element contains an open compact subgroup. I would like to explicitly construct the Haar measure on $G$. The ...
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what are the product and coproduct in the category of topological groups

I know the limits in the categories of groups, abelian groups and topological spaces and was wondering about the same thing.
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232 views

Are $\Bbb R$ and $\Bbb C$ the only connected, locally compact fields?

I heard that $\Bbb R$ and $\Bbb C$ are the only connected, locally compact fields. Does anyone know a proof for this result?
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2answers
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Why is every discrete subgroup of a Hausdorff group closed?

I have just begun to learn about topological group recently and is still not familiar with combining topology and group theory together. I have read a useful property of discrete group on the ...
9
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1answer
650 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 $A$...
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1answer
266 views

Intersection of neighborhoods of 0. Subgroup?

Repeating for my exam in commutative algebra. Let G be a topological abelian group, i.e. such that the mappings $+:G\times G \to G$ and $-:G\to G$ are continuous. Then we have the following Lemma: ...
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Why is that $\widehat{\mathbb{R}/\mathbb{Z}}\cong\mathbb{Z}$?

$\widehat{\mathbb{R}/\mathbb{Z}}\cong\mathbb{Z}$, that is, every character of $\mathbb{R}/\mathbb{Z}$ is of the form $x\mapsto e(mx)$ for some integer $m$. I was considering the dual of ${\mathbb{R}/...
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241 views

Metric on a group

Is there a non-abelian finite group $G$ with the property: If metric $d$ on $G$ is left invariant then is also right invariant?
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6answers
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Can $S^2$ be turned into a topological group?

I know that $S^1$ and $S^3$ can be turned into topological groups by considering complex multiplication and quaternion multiplication respectively, but I don't know how to prove or disprove that $S^2$ ...
27
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1answer
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Given a group $ G $, how many topological/Lie group structures does $ G $ have?

Given any abstract group $ G $, how much is known about which types of topological/Lie group structures it might have? Any abstract group $ G $ will have the structure of a discrete topological group ...
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3answers
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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 ...
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3answers
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Is a direct limit of topological groups always a topological group?

If $(G_i,f_{ij})$ is a direct system of topological groups, is it always the case that the topological\group-theoretical direct limit $G:=\varinjlim_iG_i$ is a topological group? (The topology on $G$ ...
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0answers
353 views

Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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2answers
594 views

Topological groups are completely regular

I am studying topological groups, and I have been able to do quite a lot on my own by proving the propositions in this link on my own, but when I read up wikipedia that topological groups are all ...
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3answers
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How to show that topological groups are automatically hausdorff?

On page 146, James Munkres' textbook Topology(2ed), Show that $G$(a topological group) is Hausdorff. In fact, show that if $x \neq y$, there is a neighborhood $V$ of $e$ such that $V \cdot x$ and $...
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205 views

What topological restrictions are there for a topological space to be a group?

I'm trying to provide a group structure for some Riemannian surfaces. I heard that the following result holds: Let $X$ be a compact Riemannian surface. Then $X$ admits a group structure if, and ...
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3answers
546 views

G/H is Hausdorff implies H is closed (General topology, Volume 1 by N. Bourbaki)

I am reading General topology, Volume 1 By Nicolas Bourbaki. I refer to the proof of Proposition 13. Could someone kindly explain the G/H Hausdorff $\implies$ H closed part of the proof? I ...
11
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1answer
237 views

Group Structure on $\Bbb R$

$(\Bbb R,+)$ is a topological group. Is there any other group structure on $\Bbb R$ such that it is still a topological group and this group is not isomorphic to $(\Bbb R,+)$ ? Refer to Non-...
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3answers
304 views

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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3answers
237 views

what are all the open subgroups of $(\mathbb{R},+)$

I am not able to find out what are all the open subgroups of $(\mathbb{R},+)$, open as a set in usual topology and also subgroup.
5
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2answers
354 views

Why metrizable group requires continuity of inverse?

A metrizable group is a metric space $(G,d)$ with a binary operation $\cdot$ such $(G,(\cdot))$ is a group and maps $(\cdot):G\times G\to G$ and $f:G\to G$ given by $(\cdot)(x,y)=xy$ and $f(x)=x^{-1}$ ...
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Integration over uncountable set of characters

Let $G$ be a compact (assumed Hausdorff) group and $\hat{G}$ be the set of all characters of irreducible, finite-dimensional representations of $G$. It might occur that $\hat{G}$ is uncountable. It ...
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1answer
413 views

Is there a topological group that is connected but not path-connected?

Is there a $\big($T$_0$$\hspace{-0.02 in}\big)$ topological group that is connected but not path-connected? If yes: $\quad$ Can it be complete? $\:$ (with respect to the two-sided uniform structure) ...
12
votes
3answers
748 views

Topology on the general linear group of a topological vector space

Let $K$ be a topological field. Let $V$ be a topological vector space over $K$ (if it makes things convenient, you may assume it is finite dimensional). Naive Question: Is there a canonical way of ...
11
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1answer
591 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 $\hat{G}$?...
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1answer
391 views

Completion of Topological Group with Metric

Related to this question, I'm having trouble understanding the construction of the completion of a topological group with metric structure. In particular, under what conditions is the completion also ...
2
votes
3answers
144 views

Steinhaus theorem for topological groups

$G$ is a locally compact Hausdorff topological group, $m$ is a (left) Haar measure on $X$, $A$ and $B$ are two finite positive measure in $G$, that is $m(A)>0$, $m(B)>0$. My question is: Can ...
2
votes
1answer
364 views

Finite Haar Measure if and only if Compact

This is an exercise from a book: Let $G$ be a locally compact group with Haar measure $\mu$. $\mu(\{e\})>0$ if and only if $G$ is discrete. $\mu(G)<\infty$ if and only if $G$ is compact. I ...
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1answer
85 views

Topology ad Geometry of $\mathbb{C}^n/\mathbb{Z}_k$

I would like to know topologically what the space $(\mathbb{C}^n - \{0\})/\mathbb{Z}_k$ may be thought of as. The paper I am reading says that we let $\mathbb{Z}_k$ act on $\mathbb{C}^n - \{0\}$ via ...
1
vote
1answer
250 views

Discrete subgroups of isometry group $\mathbb{R}^n$

Let $G$ be a Hausdorff topological group. We say that a subgroup $S$ of $G$ is discrete if and only if the subspace topology (from $G$) on $S$ is discrete. Note that isometry group of euclidean space ...
3
votes
1answer
125 views

Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
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1answer
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Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$ [duplicate]

Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$ where $(f\otimes g)(s) = f(s) \cdot g(s)$ where $\cdot$ is the group operation on the topological group $G. $ This is a question from ...
27
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2answers
2k views

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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Colimit of topological groups (again)

In Direct limit, Martin rightly pointed out that my naive construction (now deleted) of the colimit (direct limit) of topological abelian groups was wrong. He shows how to do it properly (at least the ...
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239 views

Non-isomorphic Group Structures on a Topological Group

Which Topological Groups Have a Unique Group Structure (up to isomorphism)? I know that there are many non-isomorphic finite groups of same order, so there are many group structures possible for ...
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1answer
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The fundamental group of a topological group is abelian [duplicate]

I want to show the fundamental group of a topological group is abelian. In fact, the question says the topological group is path connected. I do not know where I should use path-connectedness. I think,...
21
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1answer
526 views

Shrinking Group Actions

Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ ...
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549 views

Good book on topological group theory?

I'm looking for a good introduction to the theory of locally compact groups and their representations. It may assume the reader to be familiar with basic group theory, topology and measure theory.
5
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1answer
378 views

$G$ is locally compact semitopological group. There exists a neighborhood $U$ of $1$ such that $\overline{UU}$ is compact.

Let $G$ be a group and $\mathcal T$ be a locally compact topology on $G$ such that for any $a\in G$ the functions $x \mapsto ax$ and $x\mapsto xa$ are continuous on $G$. How to prove elementarily ...
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1answer
238 views

Question about Quotient space, regarding The left coset space of group $G$ with respect to a subgroup $H$

Let $G$ is a left topological group and $H$ is a subgroup of $G$. Denote by $G/H$ the set of all left cosets $aH$ of $H$ in $G$ (for each $a\in G$), and endow it with the quotient topology with ...