All Questions
Tagged with general-topology topological-groups
933 questions
0
votes
1
answer
105
views
A covering property for the discrete subgroups of a topological group
There is an example in the book "Fundamentals
of Algebraic
Topology" by Steven H. Weintraub saying that when G is a topological group and H is any discrete subgroup of G,
then the ...
- 1
-2
votes
1
answer
92
views
Are the following spaces homeomorphic to $\mathbb R^1\setminus\{0,1\}$?
I’m working through some topology questions and would like to know whether the following spaces are homeomorphic to $\mathbb R^1\setminus\{0,1\}$, which is the real line with two points removed. ...
0
votes
0
answers
40
views
Is the character group defined in $\mathbb{C}^*$ or $\mathbb{S}^1$?
I have beeen searching for the definition of the character group and depending where you read it, you may find one choice or another.
Let $G$ be a group. A character (see [Encyclopedia of Mathematics])...
- 469
6
votes
1
answer
301
views
The Coarsest Topology Which Topologizes a Group
Does there exist a coarsest topology which makes a general group $G$ a Hausdorff topological group? If so, can it be described in general, does it have a name?
In particular, I would like to show that ...
- 691
2
votes
0
answers
65
views
Finding a convergent subnet
In Bekka, de la Harpe and Valette 's book on Property (T), there is a lemma whose proof has some parts that are a bit hard for me to understand. We work with a locally compact group $G$ and $\hat G$, ...
- 21
5
votes
1
answer
204
views
Compact abelian groups with finitely many connected components
I want to look at the structure of compact abelian groups with finite number of connected components. For now, let us look at groups with exactly 2 connected components. If we let $G_0$ be the ...
- 728
1
vote
1
answer
46
views
Does the open mapping theorem hold for linear topological groups?
The open mapping theorem (of topological groups) states that a surjective continuous homomorphism of a locally compact Hausdorff group $G$ onto a locally compact Hausdorff group $H$ is an open map if $...
- 585
1
vote
1
answer
51
views
Wake topology, additive groups and Neighborhood system.
If a family $ C' $ of real-valued functions on $ X $ is an additive group, contains the constant functions, and contains the absolute value of each of its members, then the collection of sets of the ...
- 103
1
vote
1
answer
80
views
$T_0$ topological groups and countable pseudocharacter
I know that topological groups are completely regular. So, if a topological group is $T_0$, then it is $T_1$, and the pseudocharacter would be defined for the space. I was wondering about $T_0$ ...
- 1,418
1
vote
0
answers
44
views
References for topological rings and completions for a topological rings
Recently, I am reading some papers about cluster algebras and representations about quantum affine algebras. In these papers, authors often use some topological rings and completions for a topological ...
- 1,316
5
votes
0
answers
89
views
Equivalence between weak containment and convergence in Fell Topology
We first define Fell Topology. Let $G$ be a topological group and $H$ be a Hilbert space. Let $\hat G$ be the set of equivalence classes of irreducible unitary representations of $G$. We define open ...
- 357
1
vote
0
answers
31
views
Are spaces that can be embedded in a topological W-group first countable?
This question is linked to Are W-spaces with countable pseudocharacter first countable?.
Here I state the following definition:
A topological space $X$ is embeddable in a topological W-group if it is ...
- 1,418
1
vote
1
answer
194
views
Two ways to define a quotient topology. Are they equivalent?
Consider a topological group $G$ whose topology comprises open sets $U_a$.
For a subgroup $K<G$ with an induced topology assembled of the sets $K\cap U_a$, the canonical map
$$
\pi:\, G\...
- 2,078
3
votes
1
answer
77
views
Is there an Alexandrov topological group for which the topology is not discrete?
A topological space is Alexandrov if the arbitrary intersection of open sets is open. I know for a fact that there are non-discrete Alexandrov spaces, e.g. the Sierpinski space. I'm wondering however, ...
- 1,840
2
votes
0
answers
48
views
Is a compact Hausdorff group with finite homotopy type a Lie group
Suppose $G$ is a compact, Hausdorff topological group and $X$ a finite CW-complex and $X\to G$ a weak-equivalence. Does it follow that $G$ is a Lie group or could someone provide a counterexample.
In ...
- 884
3
votes
1
answer
109
views
$G$ profinite group and $H$ closed subgroup of $G$ implies $G/H$ is totally disconnected
I'm reading the proof of the theorem that if $G$ is a profinite group and $H$ one of its closed subgroups then $G/H$ is totally disconnected in Ramakrishnan and Valenza's book. The authors put aside ...
- 323
7
votes
2
answers
93
views
Compact Hausdorff Group Extension of Lie groups is Lie group
Let $K$ be a compact, Hausdorf topological group and $N\le K$ be a closed, normal subgroup. Suppose that $N$ and $L:=K/N$ are both Lie-groups. Can I conclude that $K$ admits a Lie group structure.
...
- 884
2
votes
2
answers
64
views
Fuchsian Group acts on Extended Complex Plane, but not discontinuously
I'm looking for an example of a Fuchsian group that acts discontinuously on the upper half-plane
$\mathbb{H}$, but fails to do so on the entire extended complex plane $\hat{\mathbb{C}}$ . My approach ...
- 453
1
vote
0
answers
38
views
Continuity of product of Lie group valued Holder functions
I thought the following property should be "obvious", but am having an astounding amount of trouble finding a slick, or any, proof of the following statement.
Let $g,h : [0,T] \rightarrow G$...
- 1,384
1
vote
0
answers
50
views
Continuity of the multiplication and inversion in the definition of topological group
A topological group $(G,\, *\,,\, e,\, {\cal{T}})$ is a topological space $(G,\, {\cal{T}})$ that is also a group whose unity element is $ e$ and both the multiplication
$$
\mu:\quad G \times G \...
- 2,078
2
votes
0
answers
96
views
Proof that the underlying group of the free topological group is free
I am studying Arhangel'skii and Tkachenko's book on topological groups. In the chapter on free topological groups they offer a long and involved proof that the underlying group of the free topological ...
- 61
1
vote
0
answers
67
views
Defining the completion of a group can be done only using Cauchy sequences
Let $G$ be a group, in Atiyah & MacDonald's Commutative Algebra it says that
Assume for simplicity that $0\in G$ has a countable fundamental system
of neighborhoods. The completion $\hat G$ of $G$...
- 520
1
vote
1
answer
224
views
Every simple topological group is either discrete or connected [closed]
I read the claim in a preprint that "A simple topological group is either discrete or connected". However, the explanation given was "a connected component of a topological group that ...
- 3,245
1
vote
1
answer
81
views
Generalization of the fact that for $f: S^1 \rightarrow \Bbb R$ continuous, $\exists x \in S^1 : f(x) = f(-x)$
While going through James Munkres' book on topology I came across the following problem:
Let $f: S^1 \rightarrow \Bbb R$ a continuous function. Show that there exists a point $x \in S^1$ st. $f(x)=f(-...
- 122
2
votes
0
answers
72
views
Closed subsets in Fell topology on $\mathcal{P}(G)$
I am currently working with a discrete group $G$. I am considering a set $A \subseteq \mathcal{P}(G)$ of finite subsets $F \subseteq G$ whose cardinalities are uniformly bounded. I am having troubles ...
1
vote
0
answers
54
views
Topology groups
Is there any local topological group, that cannot be obtained through taking sufficiently small neighborhood of unit in any global topological group?
2
votes
0
answers
111
views
Confusion on isomorphism profinite completion integers
Question: Prove that $\hat{\mathbb{Z}} \cong \prod_{p} \mathbb{Z}_{p}$ is an isomorphism of topological rings.
Own attempts: Although this specific question has been asked quite a few times here, ...
- 512
2
votes
1
answer
76
views
Is the inverse of an isotopy of embeddings continuous?
I was dealing with isotopies and monodromies when the following question arose, which is more subtle than it might seem at first sight.
Let $X$ and $Y$ be topological spaces. Suppose that $\Phi: [0,1] ...
- 1,361
2
votes
1
answer
77
views
Help understanding a step of the proof on the existence of a left invariant pseudometric on a topological group
The following theorem can be found in Hewitt-Ross Abstract Harmonic Analysis I, in chapter II
Theorem 8.2: Let $(U_k)_{k=1}^{\infty}$ be a sequence of symmetrics neighborhoods of $e\in G$ such that $ ...
- 117
1
vote
0
answers
42
views
Is the map $M\mapsto P$ smooth, where $P$ satisfies $PMP^{-1} = M^T$?
From this post (and other similar ones) we know, in particular, that $$ \forall M\in{\rm GL}(n,\Bbb R),\ \exists P\in{\rm GL}(n,\Bbb R)\ \text{s.t. } PMP^{-1} = M^T\ . \tag{$*$} $$
My question is ...
- 1,286
7
votes
1
answer
162
views
Are Hausdorff countably compact topological groups always normal?
A colleague and I have a result that shows that for Hausdorff countably compact W-spaces, being a topological group implies normality. But it occurred to us that (being not super experienced working ...
- 8,151
0
votes
0
answers
66
views
Algebraic Topology: Deformation Retraction and Quotient Spaces (using Mobius )
Could someone please help me with this question (and its solution)? Thanks!!
Solution Part 1 (which shows that [0,1] x {1/2} deformation retract of [0,1] x [0,1]):
I don't get how the homotopy gets ...
- 103
0
votes
0
answers
34
views
Proof by induction of $V(r) \subset V(s)$, where the Vs are neighboords of $e$ in a topological group.
Some context:
I'm trying to fill the details of a proof concerning neighborhoods indexed by dyadic rational numbers. Suppose we have a sequence $U_n$ of symmetric neighborhoods of $e \in G$ such that $...
- 117
1
vote
0
answers
57
views
Extension topology
I am reading a paper by Goldman and Sah on extension topology, but I am uncertain about the meaning of the sentences. Paraphrasing the paper:
Let $X$ be an abelian group and $M$ be a subgroup. ...
- 151
0
votes
0
answers
20
views
If $V_n$ is a cozero set, then $V_n \cap H = f^{-1}(U_n)$
I'm trying to understand Katchenko's proof about the equivalence between $z$-embedded subgroups and $\mathbb{R}$-factorizable topological groups. Just to know: A subgroup $H$ of a $\mathbb{R}-$...
- 117
2
votes
1
answer
93
views
Quotient of a compact Hausdorff and totally disconnected topological group is totally disconnected
If $G$ is a topological group which is compact Hausdorff and totally disconnected, and $H$ is a normal and closed subgroup of $G$, then Weibel's book on homological algebra claims that $G/H$ is a ...
- 313
0
votes
1
answer
41
views
Why is shift operator group action of cyclic group $\mathbb{Z}_p$ free depending whether or not $p$ is prime?
While reading one scientific article, I stumbled upon the following statement without explanation:
Let the cyclic group $\mathbb{Z}_p$ act on topological space $\left(\prod_{k=1}^p{\mathbb{R}^n}\right)...
- 329
3
votes
1
answer
56
views
Every TVS is $T_{3.5}$ (Tychonoff) even if it is not $T_0$
I'm studying the first properties of Topological Vector space, and I'm confused about the separation properties. Is every TVS $T_{3.5}$ even if it is not $T_0$? This is confirmed by this wikipedia ...
- 673
0
votes
1
answer
36
views
is $\frac{G}{N}$ second countable? [duplicate]
I'm trying to understand the proof of all compact topological groups are $\mathbb{R}-$factorizable, proof by Tkatchenko. I'm assuming all groups to be Tychonoff.
I defined the quotient group $\frac{G}{...
- 117
0
votes
0
answers
45
views
If $G$ is a topological group and $H\leqslant G$, then is $G/H$ always locally compact?
I was studying the book 'Real and Functional Analysis' by Lang, and I came across the following statement in Chapter XII (page 311):
In particular, we see that the map $\pi:G\rightarrow G/H$ is an ...
- 105
3
votes
1
answer
78
views
Regularity of the quotient of a topological group by a closed subgroup
I am trying to solve the following exercise from Chapter 2 (Exercise 7, item (d), page 146) of Munkres's Topology. The statement is as follows:
Let $H$ be a subgroup of $G$ that is closed in the ...
- 105
4
votes
2
answers
239
views
Are $O(n)$ and $SO(n)\times Z_2$ homeomorphic as topological spaces?
I'm working through Problem 4.16 in Armstrong's Basic Topology, which has the following questions:
Prove that $O(n)$ is homeomorphic to $SO(n) \times Z_2$.
Are these two isomorphic as topological ...
- 75
0
votes
0
answers
82
views
What is the universal/fine uniformity on a topological group?
I'm aware that every topological group is uniformizable: given a neighborhood $U\in\mathcal N(e)$ of the identity, the set $D_U=\{\langle x,y\rangle:x^{-1}y\in U\text{ and }xy^{-1}\in U\}$ is an ...
- 8,151
2
votes
1
answer
117
views
A locally compact Hausdorff space thas is also a group in which the operation is continuous is a topological group
I was reading the book Introduction to the topological groups by Taqdir Husain, he gives a proof for this theorem, but I had a problem understanding some part. The proof goes like this
It remains to ...
0
votes
2
answers
44
views
The inclusion $H\Gamma/\Gamma \hookrightarrow G/\Gamma$ is proper for closed subgroup $H$ and discrete subgroup $\Gamma$?
Let $G$ be a Lie group (maybe enough to assume a locally compact Hausdorff group) and $H$ be a closed subgroup and $\Gamma$ be a discrete subgroup. I wonder if the following statement is true:
The ...
- 641
2
votes
1
answer
104
views
Equivalence of Subspace and Finite Complement Topologies on a Subset
Let $X$ be a set and $\left(X, \mathcal{T}_{\mathrm{f}}\right)$ be the topological space with the finite complement topology $\mathcal{T}_{\mathrm{f}}$.
Show: For any subset $Y \subseteq X$, the ...
- 195
8
votes
2
answers
215
views
Lifting a map from the quotient by a contractible topological group
Let $X$ be a topological space, $G$ be a contractible topological group acting freely and continuously on $X$, and $Y$ be the quotient. Let $A$ be a finite-dimensional CW-complex, and $f: A \...
- 363
0
votes
0
answers
61
views
Question on Borel-measurable maps on topological groups
On page 341 of Follad's Real Analysis 2nd edition, it is written that "it is easy to see that every Borel measurable function on G is constant on the cosets of H" where $G$ is a topological ...
- 1
0
votes
0
answers
79
views
Topological groups and compact elements
I would like to ask for references about compact elements of topological groups. For the record, an element $g$ of a topological group is compact if $g$ lies in a compact subgroup.
I specifically ...
- 43
0
votes
0
answers
38
views
Orbit of closed set under compact group action
Let $X$ be a Hausdorff totally disconnected space and $G$ be a compact (or even profinite) group acting continuously on $X$. Is it true that the orbit of any closed subset $Y$ under $G$ is closed in $...
- 3,030