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

5
votes
1answer
136 views

A question about quotient under group action

Let $X$ be a Hausdorff space, and $G$ a group acting on $X$ by homeomorphisms. Let $H$ be a normal subgroup of $G$. Is it true that $X/G$ is homeomorphic to $(X/H)/(G/H)$ ? If so, can you please ...
6
votes
1answer
172 views

Question about pointwise canonically weakly pseudocompact space.

A point $x$ of a space $X$ is said to be a point of canonical weak pseudocompactness if the following condition is satisfied: For every canonical open subset $U$ of $X$ such that ...
1
vote
1answer
203 views

Is there an example of a non-orientable group manifold?

Basically what I'm looking for is a topological group that is also a non-orientable, n-dimensional manifold
4
votes
1answer
154 views

Noncommutative dual group

If $G$ is a locally compact group, we can define its dual group $\hat G$. That is set of continuous homomorphism from $G$ to circle group $\mathbb T$. My question is how to define dual group $\hat G$ ...
1
vote
1answer
82 views

Cauchy product on topological rings

Let $R$ be any commutative Hausdorff topological ring. I am looking for a preferably general condition on sequences $(x_n)_{n \in \mathbb{N}}$, $(y_n)_{n \in \mathbb{N}}$ such that the equation $$ ...
4
votes
1answer
139 views

orthogonal group of a quadratic vector space

I am reading about the orthogonal group $O(V)$ of a real finite dimensional quadratic vector space $(V,Q)$ with $Q$ nondegenerate. By definition $$O(V)=\{f:V\mapsto V |\quad Q(f(v))=Q(v) \quad \forall ...
2
votes
1answer
150 views

The intersection of open normal subgroups in a compact, totally disconnected topological group is trivial.

I am currently doing self-study on profinite groups and I'm stuck trying to prove the following lemma. If a topological group $G$ is compact and totally disconnected, then the open normal ...
5
votes
2answers
184 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?
2
votes
1answer
102 views

fundamental group of a graph

let $G$ be a connected graph and $\Omega$ its universal covering. Let $\gamma_1,\dots,\gamma_r$ be free generators of $\Gamma:=\pi_1(G)$, $v\in\Omega$ be a vertex and $s_i$ a path from $v$ to ...
1
vote
0answers
82 views

Is the Hilbert-Smith conjecture still unsolved?

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group. Is this conjecture still unsolved? Is ...
1
vote
1answer
25 views

inverse on topological semigroup

Assume $(G,\cdot)$ denotes a topological semigroup (no id, non-commutative). Let $V\subset G$ be open and take some arbitrary $g\in G$. Define $$gV^{-1}:= \bigcup_{x\in V}gx^{-1} = \bigcup_{x\in ...
5
votes
3answers
219 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
votes
1answer
102 views

If $H$ and $G/H$ are compact, then $G$ is compact.

Suppose that $G$ is a topological group and that $H$ is a subgroup of $G$ so that $H$ and $G/H$ are compact. I am trying to show that $G$ must be compact. The first idea is to use the natural map ...
1
vote
1answer
133 views

Properties of $ \text{Exp}(A) $, where $ A $ is a Banach algebra.

$ \newcommand{\Exp}{\operatorname{Exp}} $ Let $ A $ be a unital Banach algebra. For $ a \in A $, consider $$ \Exp(A) \stackrel{\text{def}}{=} \{ e^{a_{1}} e^{a_{2}} \cdots e^{a_{n}} ~|~ n \in ...
8
votes
1answer
632 views

wiki's definition of “strongly continuous group action” wrong?

Wikipedia defines strongly continuous group action as follows: A group action of a topological group G on a topological space X is said to be strongly continuous if for all x in X, the map g ↦ ...
3
votes
1answer
566 views

Topological Group, symmetric neighborhood, Hausdorff, disjoint open sets

Let $G$ be a topological group with identity $e$. If $A, B$ are subsets of $G$, we let $A * B$ denote the collection of elements $a * b$ for $a \in A, b \in B$, and we let $A^{-1}$ denote the set of ...
1
vote
2answers
80 views

Finiteness of fixed points of a Lie group action

Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$. Under which assumptions there will be a ...
1
vote
2answers
349 views

How is the general linear group a topological group?

How to see if the general linear group GL($n$), of non-singular $n$-square matrices over the real (or complex) numbers under matrix multiplication, is a topological group? How to show that matrix ...
1
vote
0answers
38 views

History: continuously differentiable groups over the real numbers

Continuously differentiable groups over the real numbers are all isomorphic to addition, as is well-known, but who proved it and when?
9
votes
0answers
129 views

Conditions for a group to admit the structure of a Lie group

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
5
votes
1answer
140 views

Action of a subgroup of finite index on a tree induced by an action of a group on a tree

Let $G$ be a group wich acts on a tree $\Gamma$. Then $U$ acts on $\Gamma$ for every $U\leq G$. Question: Why does the following hold? If $|G:U|<\infty$. Then the minimal $U$-invariant subtree ...
4
votes
1answer
196 views

Is the Haar measure of a product of finite measure and compact, finite?

Let $G$ be a locally compact group with Haar measure $ \mu $, $K \subset G$ a compact subset and $ F \subset G $ any subset of finite Haar measure $\mu (F) < \infty $. Is the Haar measure of the ...
4
votes
1answer
293 views

proof by nuke of the fact that fundamental group of topological group is abelian

"The fundamental group of a topological group is abelian". does this problem admit a proof by nuke. This is inspired by a a question in mathoverflow. The usual proof is by a Eckmann-Hilton ...
1
vote
1answer
38 views

question about $Y$-homogeneous spaces.

A subspace $Y$ of space $X$ is $h$-dense in $X$, if $Y$ is dense in $X$ and, for each $x\in X$, there exists a homeomorphism $h$ of $X$ onto itself such that $h(x)\in Y$. in this case we say that $X$ ...
3
votes
0answers
167 views

Moscow space-Examples

A space $X$ is called $Moscow$, if for each open subset $U$ of $X$, the closure of $U$ in $X$ is the union of a family of $G‎_{δ}$‎‎‎ ‎-subsets of $X$ . For example, Every first countable $T_1$ ...
5
votes
1answer
682 views

discrete normal subgroup of a connected group

could any one give me hint for this one? $G$ be a connected group, and let $H$ be a discrete normal subgroup of $G$, then we need to show $H$ is contained in the center of $G$ first of all, I have ...
0
votes
1answer
42 views

action of $O(n,\mathbb{R})$ on $S^n$

I need to know what is the action of $O(n,\mathbb{R})$ on $S^n$, and $O(n,\mathbb{R})/O(n-1,\mathbb{R})\cong S^{n-1}$, how does $O(n-1,\mathbb{R})$ sit inside $O(n,\mathbb{R})$? The obvious action ...
0
votes
1answer
283 views

Continuous homomorphism into locally compact Hausdorff group

Could any one give me hint to solve this one? $f:G\rightarrow H$ is continuous homomorphism into a locally compact Hausdorff group $H$. Then we need to show $f$ is necessarily open. all spaces 2nd ...
3
votes
2answers
150 views

why the orthogonal group $O(k,l)$ is homotopy equivalent to $SO(K)\times SO(l)$

I want to prove that the orthogonal group $O(k,l)$ (http://en.wikipedia.org/wiki/Indefinite_orthogonal_group)is homotopy equivalent to $SO(k)\times SO(l)$, so that ...
7
votes
1answer
94 views

topological group operation vs homotopy group operation

Let $X$ be a topological group. Let $\tau_1$ and $\tau_2$ representing elements of $\pi_n(X)$. Is it true that $$ [\tau_1] [\tau_2] = [\tau_1 \tau_2] $$ in $\pi_n(X)$?, where of course "$[\tau_1] ...
3
votes
1answer
298 views

Metrizable group

Let $ G $ be a metrizable group. If (i) $ K $ is a closed normal subgroup of $ G $ and (ii) both $ K $ and $ G/K $ are complete, then $ G $ is complete. Here is how I am proceeding: It can be ...
1
vote
1answer
114 views

A new group operation on the fudamental group [duplicate]

Suppose $G$ is a topological group with operation $\cdot$ and identity element $x_0$. Let $\Omega (G, x_0)$ denote the set of all loops in $G$ based at $x_0$. For $f, g\in\Omega (G, x_0)$ define ...
1
vote
1answer
199 views

A couple of questions about closed subgroups of a topological group.

I am reviewing my previous exams, and I completely missed the following two-part question. It deals with closed subgroups of topological groups under certain situations. I am having trouble working it ...
3
votes
2answers
100 views

Why are topological groups semitopological groups?

As part of a casual self-study of topology, I have started fooling around with topological groups. I noticed that the article on Wikipedia mentioned that "weakening the continuity conditions" gives ...
5
votes
1answer
721 views

Quotient of a locally compact Hausdorff space by a proper action is Hausdorff

I am trying to prove the following: Let $G$ be a topological group acting properly on a Hausdorff locally compact space $X$, i.e. preimages of compacts sets by the map $$G\times X\to X\times ...
1
vote
1answer
64 views

Action and bounded orbits

Let $H$ be an open group such that, $H$ act continously an by isometrieson a metric space $(X,d)$ ($\forall h\in H$, the map $X\ni x\longmapsto h.x\in X$ is an isometry.). Recall that for $x_{0}\in X$ ...
4
votes
2answers
119 views

Continuous action

Let $G$ be a polish group, $H$ an open subgroup of $G$. Now assume that $H$ acts by isometries (For all $h\in H$, the map $X\ni x\longmapsto X$ is an isometry) and continously on a metric space ...
15
votes
3answers
1k views

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 ...
2
votes
1answer
152 views

On an existence of a quasi-finite left- invariant Borel measure in a non-locally compact Polish group

Let $(G,B(G))$ be a Polish group. A Borel set $A \subset G$ is called Haar null if there is a Borel probability measure $\mu$ in $G$ such that $\mu(g(A))=0$ for each $g \in G$. A Borel measure ...
3
votes
0answers
71 views

Induced action of topological groups

Let $G$ be a polish group, $H$ be an open subgroup of $G$ and $X$ be any metric space on which $G$ act. I want to show the following fact: If the restriction to $H $of the action of $G$ on $X$ is ...
0
votes
1answer
260 views

Stone-Cech compactification

Is the following statement true or not? A locally compact Hausdorff space $X$ is a group if and only if its Stone-Cech compactification $\beta X$ is a group. Thanks.
10
votes
1answer
271 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
votes
1answer
106 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
145 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 ...
22
votes
1answer
786 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
418 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
415 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
123 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
votes
0answers
115 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 ...