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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269 views

Is $[0,1]$ a topological group?

Can one endow the unit interval $[0,1]$ with a group operation to make it a topological group under its natural Euclidean topology?
7
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1answer
104 views

Closed subgroups of n copies of the p-adic integers

What do closed subgroups of $\mathbb{Z}_p \oplus \cdots \oplus \mathbb{Z}_p$ look like (where there are $n$ summands in the direct sum)?
3
votes
2answers
142 views

Is Hom$(G,-)$ left exact if morphisms are required to be continuous?

Suppose now that the objects in question are abelian topological groups $G$ so that morphisms are continuous group homomorphisms. Given an exact sequence of abelian topological groups $0 \to G''\to G ...
20
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1answer
741 views

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 ...
3
votes
1answer
407 views

Difference between the SU(2) and SO(3) lie groups and their lie algebras

In many places I have seen the SU(2) and SO(3) lie algebras used interchangeably. How are they exactly identical? Moreover, what about their lie groups? Are they identical as well. It would be great ...
1
vote
1answer
65 views

$\overline H$ is a normal subgroup of a topological group $G$.

Let $G$ be a topological group. How can we prove that if $H$ is a normal subgroup of $G$, then $\overline H$ is a normal subgroup of $G$ also? First of all, we have to prove that $\overline H$ is a ...
3
votes
1answer
402 views

Is the orbit space of a Hausdorff space by a compact Hausdorff group Hausdorff?

Let $G$ be a compact Hausdorff group. Let $X$ be a Hausdorff space. Suppose $G$ acts continuously on $X$. Is the orbit space $X/G$ Hausdorff? If not, I would like to know an counter-example. Remark ...
1
vote
1answer
122 views

Gillman-Jerison Theorem

How can i prove it? [Gillman and Jerison] If a dense subspace $Y$ of a Tychonoff space $X$ is $C-embedded$ in X, then $Y$ is $ G‎‎_{\delta‎‎‎}-dense‎ $‎ in $X$.
3
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0answers
113 views

Group theoretic lemma about the extension of homomorphisms of profinite groups

I have a question about a group-theoretic lemma proven in Galois Groups and Fundamental Groups by Tamas Szamuely. Suppose we have a profinite group $\Gamma$, a closed normal subgroup $N \subset ...
3
votes
1answer
203 views

discrete subgroup of locally compact abelian group

Let $G$ be a locally compact abelian infinite group but non-compact. In some paper, the author claims that the dual group $\widehat{G}$ contains an infinite discrete group $K$. What do you think ...
2
votes
1answer
161 views

In topological groups. Is every neighborhood of $e$ supset of a square of a symmetric neighborhood of $e$?

Let $G$ be a topological group, $U$ is a neighborhood of $e$ which is the unit element of $G$. My question is does there exist a neighborhood $H \subseteq U$ of $e$ s.t. $H^{-1}=H$ $H\cdot ...
1
vote
1answer
70 views

Conjugacy classes of a compact matrix group

Let $G$ be a compact matrix group. May I know why the conjugacy classes of $G$ is necessarily closed? I tried to argue by taking limits but to no avail so is there a hint on how to tackle this ...
0
votes
1answer
97 views

In topological groups. Is every neighborhood of $e$ supset of a neighborhood subgroup?

Let $G$ be a topological group, $U$ is a neighborhood of $e$ which is the unit element of $G$. My question is does there exist a neighborhood $H \subseteq U$ of $e$ s.t. $H$ is a subgroup of $G$? ...
2
votes
2answers
52 views

Is operator open at topological groups?

Let $(G,\cdot,\mathscr{T})$ be a topological group, then $\cdot$ is indeed continuous, but is it open(close) mapping? It is true at $(\mathbb R,+,\mathscr{T}_{Ord})$, so I guess it is also true in ...
1
vote
1answer
126 views

How to conclude that a path is non-trivial element of $\pi_1(M)$

Let $M^3$ be a compact manifold. If $\mathbb{RP}^2$ is embedded in $M$. Suppose, by contradiction, that $i_\sharp: \pi_1(\mathbb{RP}^2) \longrightarrow \pi_1(M)$ is non-injective and that the normal ...
2
votes
2answers
157 views

Open subgroups of a topological group are closed

Let $G$ be a topological group such that for each $x \in G$ the mapping $x\mapsto xy$ is a homeomorphism. If $H$ is a open subgroup of $G$, prove that $H$ is also closed Could anyone just give ...
1
vote
1answer
140 views

topological groups basic facts

How to show that the usual metric with the usual addition is a topological group? Can anybody please explain me briefly about topological groups and the way that I need to approach to this question?
6
votes
1answer
249 views

How does Pontryagin duality fit into the general cohomology theory framework?

Pontryagin duality implies the isomorphic relation of the function space $C(G)$ on a locally compact group $G$ to the function space on it's dual group $\hat G \overset{\sim}{=}\text{Hom}(G,T)$, ...
2
votes
1answer
71 views

Dense subalgebras of topological algebras

Let $A$ be a topological unital algebra and let $B$ be its dense subalgebra with unit. Let $I$ be a right ideal of $B$. Is the closure of $I$ a right ideal of $A$?
3
votes
1answer
456 views

Normal subgroups of the Special Linear Group

What is some normal subgroups of SL(2, R)? I tried to check SO(2, R), UT(2, R), linear algebraic group and some scalar and diagonal matrices, but still couldn't come up with any. So can anyone give ...
1
vote
1answer
338 views

Universal cover of complete hyperbolic surfaces and torsion-free, discrete groups of isometries of $\mathbb{H}^2$

I'm taking a course this semester, and in it we proved that any complete hyperbolic surface is universally covered by $\mathbb{H}^2$. The text, found at ...
3
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1answer
175 views

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.
2
votes
1answer
144 views

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 $properly\;discontinuous$ ...
0
votes
0answers
96 views

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 ...
0
votes
1answer
68 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 ...
15
votes
2answers
578 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 ...
6
votes
1answer
103 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 ...
4
votes
1answer
159 views

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 ...
7
votes
1answer
382 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 ...
4
votes
2answers
158 views

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 ...
0
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1answer
138 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. ...
2
votes
4answers
310 views

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, ...
4
votes
1answer
151 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.
3
votes
2answers
55 views

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;\; ...
1
vote
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
votes
1answer
140 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
votes
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
votes
1answer
137 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 ...
3
votes
2answers
420 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 ...
4
votes
1answer
256 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} \: = \: ...
9
votes
1answer
448 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 ...
4
votes
1answer
176 views

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 ...
4
votes
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 ...
4
votes
1answer
915 views

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 $$ ...
22
votes
2answers
1k 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 ...
3
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0answers
298 views

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 ...
7
votes
3answers
2k views

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
votes
1answer
132 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 ...
2
votes
1answer
268 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. ...
6
votes
1answer
110 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 ...