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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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 ...
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606 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 ↦ ...
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551 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 ...
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79 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 ...
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335 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 ...
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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?
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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 ...
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139 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 ...
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190 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 ...
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291 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 ...
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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 $X$ ...
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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 $T_1$ ...
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654 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 ...
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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 ...
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280 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 ...
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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 ...
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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] ...
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295 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 ...
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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 ...
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197 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 ...
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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 ...
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720 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 ...
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63 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 $x_{0}\in X$ ...
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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 ...
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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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1answer
151 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 ...
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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 ...
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255 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.
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270 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?
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105 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)?
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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 ...
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765 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 ...
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412 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 ...
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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 ...
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408 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 ...
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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 $X$.
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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 ...
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206 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 ...
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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 ...
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76 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 ...
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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 $G$? ...
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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 ...
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127 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 ...
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2answers
178 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 ...
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141 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?
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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 $\hat G \overset{\sim}{=}\text{Hom}(G,T)$, ...
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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 $A$?
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471 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 ...
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362 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 ...
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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.