All Questions
54 questions
0
votes
1
answer
98
views
Is every subgroup closed in this complete, nondiscrete topological group?
Another question on Mathoverflow (here: Complete topological groups in which all subgroups are closed) asks if there exists a complete, nondiscrete topological group $G$ such that all subgroups of $G$...
- 361
5
votes
1
answer
251
views
In a topological group, is $G/A\to G/B$ a covering map if $A$ is open in $B$?
Let $G$ be a (Hausdorff) topological group, let $A,B$ be closed subgroups of $G$ such that $A$ is an open subgroup in $B$. Then we have an open continuous map $f:G/A\to G/B$, with typical fiber $B/A$. ...
- 658
0
votes
0
answers
96
views
Idempotent conjecture and (weak) connectivity of (a reasonable) dual group
What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?
The Motivation: The motivation comes from the idempotent conjecture of ...
- 356
5
votes
1
answer
287
views
Extreme amenability of topological groups and invariant means
Recently I'm reading the paper Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups by Pestov. When it comes to the definition of an extremely amenable topological group, it ...
- 225
0
votes
0
answers
123
views
Classification of closures of additive subgroups of $\mathbb{R}^n$
If $G$ is an additive subgroup of the real numbers $\mathbb{R}$ and $\overline{G}$ is the topological closure of $G$ then either
$\overline{G} = a \cdot \mathbb{Z}$ for some $a \in \mathbb{R}$, or
$\...
- 2,221
0
votes
0
answers
152
views
Left-side cosets of an open subgroup
Let $G$ be a topological group and $H$ its closed subgroup. $K$ and $L$ are open subgroups of $G$ and $H$ respectively. Let $g_{1}, g_{2}\in G$. We assume $L\cap g_{1}K\neq \emptyset$ and $L\cap g_{2}...
- 479
8
votes
1
answer
509
views
About locally compact groups without compact subgroups
Is every Hausdorff, locally compact group that does not contain any non-trivial compact group, finitely dimensional?
- 4,786
9
votes
2
answers
902
views
Are locally compact, Hausdorff, locally path-connected topological groups locally Euclidean?
Is every locally compact, Hausdorff, locally path-connected topological group $G$ locally Euclidean? (That would imply of course also being a Lie group.) Is it true when countable basis is assumed? I ...
- 2,390
11
votes
1
answer
991
views
Why are homeomorphism groups important?
For a compact metric space $X$ let $\mathcal H(X)$ denote the set of homeomorphisms in the compact-open topology (also generated by sup metric). It is known that $\mathcal H(X)$ is a Polish ...
- 3,317
-2
votes
1
answer
131
views
$G$- space is locally compact [closed]
Suppose $X$ is a topological space ,$G$ Is a locally compact group.If the quotient space $G\backslash X$ is compact,can we deduce that $X$ is locally compact?
- 451
2
votes
1
answer
82
views
Structure of extensions arising in Lie approximation of connected groups
My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known:
Let $G$ be a connected, locally compact, Hausdorff group, ...
- 25.8k
5
votes
0
answers
316
views
Polish groups with no small subgroups
Definitions.
A Polish group is a topological group $G$ that is homeomorphic to a separable complete metric space.
A group $G$ has no small subgroups if there exists a neighborhood $U$ of the identity ...
- 998
5
votes
2
answers
328
views
Set of topologies on a group making it a compact Hausdorff topological group
Maybe stupid, but from the following well known facts about compact Hausdorff (CH) spaces:
CH topologies on a given set are pairwise incomparible (one is not finer or coarser than the other).
There ...
- 1,031
12
votes
0
answers
172
views
A connected Borel subgroup of the plane
It is known that the complex plane $\mathbb C$ contain dense connected (additive) subgroups with dense complement but each dense path-connected subgroup of $\mathbb C$ necessarily coincides with $\...
- 41.8k
1
vote
0
answers
121
views
A section over an orbit space
Let $G$ be a compact second countable Hausdorff group, and let $X=G/H$ be a homogeneous space with $H\subset G$ a closed subgroup. Let further $K\subset G$ be another closed subgroup.
Questions:
...
- 1,959
2
votes
1
answer
76
views
Haar-$\mathcal{I}$ set and Polish groups
Let $\mathcal{I}$ be a semi-ideal of sets with empty interior on a compact metrizable space $K$. Let an $F_σ$-set $A$ in a Polish group $X$ generically Haar-$\mathcal{I}$.
Then is $A$ always ...
- 1,755
2
votes
1
answer
153
views
Every quasicharacter of an open subgroup extends to a quasicharacter on the whole group
Let $H$ be an open subgroup of a locally compact Hausdorff abelian group $G$. Assume that $G/H$ is a finitely generated abelian group. Let $\chi: H \rightarrow \mathbb{C}^{\ast}$ be a continuous ...
- 6,170
14
votes
1
answer
295
views
Is $Alt_\omega$ a dense subgroup of a non-discrete locally compact topological group?
Let $S_\omega$ be the group of bijections of the countable ordinal $\omega:=\{0,1,2,\dots\}$ and $Alt_\omega$ be the subgroup of $S_\omega$ consisting of even permutations of $\omega$ (i.e., the ...
- 41.8k
5
votes
2
answers
454
views
Is each locally compact group topology on the permutation group discrete?
Question. Is each locally compact group topology on the permutation group $S_\omega$ discrete?
Here $S_\omega$ is the group of all bijections of the countable ordinal $\omega$. A group topology on a ...
- 41.8k
1
vote
0
answers
109
views
Toral subgroup acting regularly on the homogeneous space
Let $G$ be a connected second countable compact Hausdorff group, and let $H\subset G$ be a closed subgroup such that the homogeneous space $G/H$ has continuum cardinality. For every $x\in G/H$ let $...
- 1,959
11
votes
2
answers
578
views
Homeomorphisms vs Borel automorphisms
Let $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ be the groups of homeomorphic and Borel automorphisms of a space $M$, respectively.
Question: Are $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ ...
- 1,959
4
votes
1
answer
348
views
Is there a topologizable group admitting only Raikov-complete group topologies?
Definition. A group $G$ is called complete (resp. non-topologizable) if each Hausdorff group topology on $G$ is Raikov-complete (resp. discrete). It is clear that each non-topologizable group is ...
- 41.8k
2
votes
0
answers
82
views
Uniquely divisible neighborhoods of identity in topological groups
Let $G$ be a (finite dimensional real) Lie group, and let $A\subset G$ be an open neighborhood of identity. If $A=\operatorname{Exp}(\mathcal{A})$ is the injective range of the exponential map from a ...
- 1,959
4
votes
1
answer
328
views
Is the Cantor set countable dense homogeneous in pairs?
I know that the Cantor set is countable dense homogeneous. My question is: if A,B,C,D are countable dense subsets of the Cantor set such that the pairs A and B and C and D are disjoint, there exists a ...
- 173
8
votes
0
answers
569
views
example of an n-transitive but not infinitely transitive group action on a space
Definition. An action of a group $G$ on a set $X$ is strongly $n$-transitive if $G$ acts transitively on $n$-tuples of distinct elements in $X$ (via the diagonal action), and is $n$-transitive if $G$ ...
- 4,588
2
votes
0
answers
139
views
Centralizer of a dense subgroup in a maximal subgroup of a reductive group
I am looking for a reference to the following statement
"Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$...
- 21
4
votes
0
answers
87
views
Almost invariance in compact quotients of locally compact groups
While trying to get an analogue of Weiss's monotiling result for amenable residually finite groups
in the topological setting, I face the following problem.
Let $G$ be a locally compact amenable ...
8
votes
1
answer
229
views
Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups
An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a ...
- 10.5k
13
votes
1
answer
459
views
A generalization of residual finiteness to topological groups
Consider the following generalization of residual finiteness to
topological groups.
A locally compact Hausdorff group $G$ is called residually compact if
for every compact $K \subseteq G$ there is a ...
13
votes
0
answers
421
views
A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?
Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
- 41.8k
0
votes
1
answer
149
views
Can every non-discrete topological group G be algebraically generated by a nowhere dense subset?
Is there somone help me to show that if this problem have positive Answer :
Problem :Can every non-discrete topological group G be algebraically gen-
erated by a nowhere dense subset ?
Thank ...
9
votes
1
answer
401
views
Meager subgroups of compact groups
Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre.
Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...
- 1,338
4
votes
0
answers
90
views
Topological systems of imprimitivity
Let $G$ be a group acting by homeomorphisms on a topological space $X$. $G$ is topologically transitive if every open $G$-invariant subset of $X$ is empty or dense.
Here is an attempt to define ...
- 4,728
4
votes
1
answer
292
views
Can an abelian group have a minimal group topology?
In the abstract of this paper, it is said that a minimal group topology on an abelian group is not Hausdorff.
Suppose $G$ is an abelian group and $\mathcal T$ is a minimal group topology on $G$ and ...
- 1,145
1
vote
3
answers
172
views
Is the poset of all precompact group topologies on an abelian group $G$, order-isomorphic to $\operatorname{Sub}(\hat{G})$?
In this page, in abstract, it is claimed that the poset of all Hausdorff precompact group topologies on an abelian group $G$, is order-isomorphic to the the subgroup lattice of $\hat{G}$, the ...
- 1,145
2
votes
0
answers
104
views
Selecting dense diagonals in $\Bbb T^2$
Let $p$ be a prime number and let $G=\bigcup_{n\in \Bbb N}\{\exp(k\frac{2\pi i}{p^n})\mid k\in \Bbb Z\}$ be a Prüfer group. For homomorphisms $f,g:G\to G$ let $H_{f,g}=\{(f(x),g(x))\mid x\in G\}$. ...
- 1,145
14
votes
1
answer
1k
views
Distributivity of group topologies on $\Bbb Z$
Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$.
It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice [1].
Is $(\mathcal L,\subseteq)$ distributive?
$$~$$
[1] ...
- 1,145
7
votes
1
answer
455
views
Group structure on an arbitrary completely regular topological space that makes $(x,y)\mapsto xy^{-1}$ continuous at $(1,1)$
Let $(G,\mathcal T)$ be a completely regular topological space. Is there a group structure on $G$ such that the function
$$f:G\times G\to G$$
$$f(x,y)=xy^{-1}$$
is continuous at $(1,1)$?
- 1,145
4
votes
1
answer
239
views
Are infinite groups "locally topologizable"?
Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point?
The question is inspired by and related to this ...
- 3,932
32
votes
1
answer
2k
views
A group allowing exactly 7 group topologies
Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{\text{trivial}}, \mathcal T_{\text{discrete}}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with
$$...
- 1,145
5
votes
0
answers
93
views
Separation of topological group elements by invariant neighbourhooods
Let $G$ be a topological group that is Hausdorff, that is, for every pair $(g,h)$ of distinct elements of $G$, there exist disjoint open sets $U_g$ and $U_h$ such that $g \in U_g$ and $h \in U_h$.
...
- 4,728
6
votes
1
answer
338
views
Topological groups defined by completely disconnected subgroups
Can you define a group topology on a group by specifying which subgroups should be discrete with respect to that topology (where a subgroup $S$ of $G$ is discrete if each $s\in S$ has an open ...
- 345
0
votes
1
answer
111
views
Is a weakly separable group always Lindelöf?
By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...
- 135
15
votes
1
answer
986
views
Is a left topological group which is a manifold a topological group?
Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not ...
- 1,269
13
votes
2
answers
514
views
subsets of groups which have to be closed no matter what
One example of a subset of a group $G$ which has to be closed in any topology on $G$ compatible with the group operations is a centraliser. Are there any other interesting examples?
- 2,125
12
votes
1
answer
746
views
Which topological spaces are coset spaces of locally compact groups?
What is a topological characterization of the class of spaces that have the form $G/H$ for a locally compact, Hausdorff group $G$ and a closed subgroup $H$ ?
Such a space $X=G/H$ necessarily ...
- 3,497
1
vote
0
answers
430
views
Intersection of cocompact closed normal subgroups
Let $G$ be a locally compact Hausdorff topological group.
Definition A closed normal subgroup $H \unlhd G$ is called cocompact if $G/H$ is compact with respect to the quotient topology.
Note that ...
- 1,498
12
votes
2
answers
741
views
Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group? [xpost from math.SE]
And what else can be said, if so?
(Original math.SE post)
In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (...
- 2,585
5
votes
1
answer
2k
views
Proof that the Pontryagin dual of a topological group is a topological group
I'm looking for a proof that the Pontryagin dual $G^*$ of a topological group $G$ is a topological group.
It's very easy to prove that $G^*$ is a group, my troubles are in proving that the map $G^* \...
- 285
11
votes
2
answers
2k
views
Two Definitions of "Character" of topological groups
When I first met the concept of "characters" of topological groups in Pontryagin's book "Topological groups", it was defined as follows:
Let $G$ be a topological group. A ...
- 945