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.

learn more… | top users | synonyms

4
votes
1answer
929 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 $$ ...
9
votes
2answers
1k views

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 ...
14
votes
6answers
546 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 ...
6
votes
2answers
1k views

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 ...
7
votes
1answer
572 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 ...
7
votes
1answer
777 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 ...
10
votes
2answers
1k views

Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

Could you please explain me the reason why they are isomorphic? Thanks, bye!
5
votes
3answers
217 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.
10
votes
1answer
730 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 $\mathcal{L}^1(G, \mu)$. ...
15
votes
2answers
596 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 ...
15
votes
2answers
912 views

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.
8
votes
2answers
928 views

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 ...
7
votes
1answer
231 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: ...
9
votes
1answer
457 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 ...
7
votes
1answer
500 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 ...
4
votes
2answers
196 views

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 ...
4
votes
1answer
219 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?
15
votes
3answers
929 views

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 $G$ ...
10
votes
2answers
463 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 ...
7
votes
2answers
162 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 ...
5
votes
2answers
176 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?
7
votes
3answers
256 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 ...
5
votes
2answers
318 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 $f(x)=x^{-1}$ ...
4
votes
1answer
350 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 ...
3
votes
0answers
73 views

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 ...
3
votes
2answers
349 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 ...
2
votes
3answers
109 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 ...
1
vote
1answer
326 views

$G$ topological group, $H$ discrete normal subgroup, $p$ projection, form Covering Space.

Let $G$ be a topological group. Let $H$ be a discrete normal subgroup of $G$. Let $p : G \to G/H$ be the projection map. Show that $(G, p, G/H)$ form a covering space. Here is what I have so far: ...
3
votes
0answers
96 views

Prove that the quotient map $P:G \to G/H$ is a covering space.

Let $G$ be a topological group and $H$ is a subgroup of $G$. Suppose that the subspace topology on $H$ is the discrete topology. Prove that the quotient map $P:G \to G/H$ is a covering space. My ...
3
votes
1answer
96 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 ...
0
votes
1answer
115 views

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 ...
22
votes
1answer
764 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 ...
12
votes
2answers
194 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 ...
18
votes
3answers
928 views

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 ...
7
votes
2answers
397 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.
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 ...
5
votes
1answer
329 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 ...
4
votes
1answer
820 views

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 ...
21
votes
1answer
492 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 $\rho : G\times X\rightarrow X$ ...
5
votes
2answers
200 views

equation involving the integral of the modular function of a topological group

Let $G$ be a locally compact topological group and $H$ a closed subgroup. Choose a left Haar measure $d\zeta$ for $H$, and let $d\mu$ be any measure for $G$. Also let $f$ and $g$ be continuous ...
10
votes
1answer
177 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 ...
9
votes
1answer
171 views

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 ...
8
votes
1answer
378 views

An equivalent definition of the profinite group

A profinite group is by defination a topological group $G$ which is Hausdorff , compact and totally disconnected. How to prove the following equivalent defination: A compact Hausdorff group is ...
5
votes
1answer
167 views

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. ...
4
votes
0answers
175 views

How many compatible group structures does a topological space admit?

Suppose I have a topological space $(G,\tau)$ and am interested in whether there exists a topological group $(G,*,\tau)$. In other words, can we assign a binary operation $*$ to $(G,\tau)$ which ...
12
votes
1answer
314 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) ...
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 ...
6
votes
1answer
115 views

Uniqueness of compact topology for a group

Suppose $G$ is a compact $T_2$ group. Can there be other compact $T_2$ topologies on $G$ which also turn $G$ into a topological group? ($T_2$ refers to the Hausdorff separation axiom)
5
votes
1answer
299 views

Product of totally disconnected space is totally disconnected?

I read that the cartesian product with the product topology of a family of totally disconnected topological spaces is totally disconnected, too. Is that true? How are the connected components in the ...
3
votes
1answer
546 views

proving that $SO(n)$ is path connected

Our professor gave us exercise to show that $G=SO(n,\mathbb R)$ is path connected. He gave some hints, using them I have come upto this far: I have shown that $SO(n)$ acts on $S^{n-1}$ transitively ...