All Questions
Tagged with gn.general-topology topological-groups
213 questions
5
votes
2
answers
109
views
A reasonable topology on the group of minimal usco maps
An usco map is an abreviation for an upper semicontinuous multi-valued map with non-empty compact values. An usco map $f:X\multimap \mathbb R$ is called minimal is it coincides with each usco map $g:X\...
- 4,727
12
votes
1
answer
321
views
If $G$ is a paracompact topological group, then is $G \times G$ paracompact?
If $G$ is a paracompact topological group, then is $G \times G$ paracompact?
This question is raised by Gepner and Henriques (first paragraph of 2.2). Of course, this is not true for arbitrary ...
- 18.6k
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 $\...
- 41.9k
3
votes
1
answer
175
views
Has the Erdős space the structure of a monothetic topological group?
This question is motivated by this MO-problem asking if the Erdős spaces $\mathfrak E$ and $\mathfrak E_c$ admit a self-homeomorphism with dense orbits of points.
The affirmative answer would follow ...
- 6,734
3
votes
1
answer
184
views
Can approximately periodic functions be perturbed to periodic functions on a locally compact group?
Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...
- 1,023
3
votes
1
answer
152
views
A question on quasitopological group
Suppose that $G$ is a regular feebly compact Moore quasitopological group. Must $G$ be a topological group? This was previously posted here on MathSE also.
A semitopological group $G$ is a group $G$ ...
- 4,713
10
votes
1
answer
366
views
Are all compact subsets of Banach spaces small in a measure-theoretic sense?
Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset ...
- 18.4k
3
votes
1
answer
201
views
Is each closed subgroups of $\mathbb R^\omega$ isomorphic to a Tychonoff product of locally compact Abelian groups?
It is known that any closed linear subspace of $\mathbb R^\omega$ is topologically isomorphic to $\mathbb R^n$ for some $n\in\omega$.
Problem 1. Is each closed subgroup of $\mathbb Z^\omega$ (or ...
- 11.1k
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:
...
- 11.6k
3
votes
1
answer
160
views
Distance for $GL_n(\mathbb{R})/GL_n(\mathbb{Z})$
One can define the convergence of a sequence $(\Lambda_k)_k$ of full rank lattices as folow : $(\Lambda_k)\underset{k\rightarrow +\infty}{\longrightarrow} \Lambda \iff \forall k\in \mathbb{N} ,\exists ...
- 11.6k
2
votes
1
answer
76
views
Haar-$\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 ...
- 41.9k
2
votes
0
answers
81
views
If $H$ and $Z$ are closed subgroups generating $G$, is $H \times Z \rightarrow G$ an open map?
Let $G$ be a Hausdorff topological group with center $Z$ and closed subgroup $H$. Suppose that $H.Z = G$. Is the product map
$$H \times Z \rightarrow G$$
necessarily an open map? That is, can we ...
- 6,180
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 ...
- 41.9k
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 ...
- 63.9k
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
2
votes
0
answers
64
views
On minimality of semitopological and quasitopological groups
The phenomemnon of minimality is well-studied in the realm of topological groups.
Let us recall that a topological group $X$ is minimal if each bijecive continuous homomorphism $h:X\to Y$ to a ...
- 41.9k
2
votes
0
answers
102
views
Is this concrete set generically Haar-null?
This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete.
First we recall the definition of a generically Haar-null set in ...
- 41.9k
5
votes
0
answers
214
views
On generically Haar-null sets in the real line
First some definitions.
For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...
- 41.9k
7
votes
1
answer
207
views
The square of a ccc topological group
Jensen proved that under $\Diamond$ there is a homogeneous Suslin continuum, so the square of a ccc homogeneous space can fail to be ccc. What about ccc topological groups?
Is there a ccc ...
- 11.5k
24
votes
4
answers
7k
views
Compact open topology
What is the intuition behind using compact open topology for eg. in the case of Pontryagin dual ?
- 4,713
17
votes
6
answers
6k
views
What is a good book on topological groups?
I am looking for a good book on Topological Groups. I have read Pontryagin myself, and I looked some other in the library but they all seem to go in length into some esoteric topics.
I would love ...
- 4,713
4
votes
0
answers
133
views
Equivalence of Flat Fiber Bundles vs Equivalence of Group Actions on the Fiber
Let's consider all flat fiber bundles with base space $B$ and fiber $F$, where $B$ and $F$ are compact and at least CW-complexes. (perhaps even topological/smooth manifolds if that helps)
All those ...
- 404
3
votes
4
answers
1k
views
Topologically split extensions of topological groups
Let $1 \to N \to G \to H \to 1$ be a short exact sequence of topological groups. Such an exact sequence is said to be topologically split if $G$ is $N \times H$ as a
topological space.
Can someone ...
- 4,713
6
votes
1
answer
618
views
Is every compact monothetic group metrizable?
If $G$ is a compact (Hausdorff) topological group with a dense cyclic subgroup, is it necessarily true that $G$ is first countable? This claim seems to be implicit in a paper that I am reading at the ...
- 4,713
2
votes
0
answers
49
views
Does each weakly feathered topological group admit an injective homomorphism into a feathered topological group?
A topological group $G$ is called
$\omega$-$\mathit{narrow}$ if for each non-empty open set $U\subset G$ there exists a countable subset $C\subset G$ such that $G=CU=UC$;
$\mathit{feathered}$ if $...
- 41.9k
1
vote
0
answers
128
views
The group of polynomial homeomorphism of the plane
Let $G$ be the set of all homeomorphisms $f$ of $\mathbb{R}^2$ such that
both $f$ and $f^{-1}$ are polynomial maps.
We equip $G$ with the compact open topology and the obvious group ...
- 356
4
votes
1
answer
147
views
Equicontinuity and orbits of compact open sets
Let $X$ be a compact zero-dimensional space, let $S \subseteq \mathrm{Homeo}(X)$ and let $U$ be a compact open subset of $X$. Suppose that $s^{-1} \in S$ for all $s \in S$, and that $S$ restricts to ...
- 4,728
7
votes
0
answers
214
views
Is each completely minimal topological group minimal?
A topological group $G$ is called
$\bullet$ minimal if it admits no strictly weaker Hausdorff group topology;
$\bullet$ completely minimal if it is Raikov-complete in each weaker Hausdorff group ...
- 41.9k
9
votes
3
answers
952
views
Is there a non-trivial topological group structure of $\mathbb{Z}$?
More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?
- 5,409
1
vote
0
answers
47
views
Is the minimality of complete topological groups recognizable by closed separable subgroups?
A topological group is called minimal if it admits no strictly weaker Hausdorff group topology.
By Prodanov-Stoyanov Theorem, a complete Abelian topological group is minimal if and only if it is ...
- 41.9k
3
votes
0
answers
143
views
Is an Abelian topological group compact if it is complete and Bohr-compact?
A topological group $G$ will be called Bohr-compact if its Bohr topology (i.e., the largest precompact group topology) is compact and Hausdorff.
A topological group $G$ is Bohr-compact if it admits ...
- 356
1
vote
0
answers
206
views
A reasonable topology on the automorphism group of an $\omega$-narrow topological group?
For a topological group $X$ by $Aut(X)$ denote the group of topological isomorphisms $h:X\to X$. If $X$ is compact then the compact-open topology turns $Aut(X)$ into an $\omega$-narrow topological ...
- 41.9k
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 ...
- 41.9k
2
votes
0
answers
73
views
Is there a star Lindelöf topological group which is not star countable?
I'm interested in this question:
Is there a star Lindelöf topological group which is not star countable?
A topological space $X$ is said to be star countable if whenever $\mathscr{U}$ is an open ...
CommunityBot
- 1
1
vote
0
answers
149
views
Groups and non-trivial finite topologies
Recently, I have been "updating" myself in the field of topological groups, and, in doing this, I remembered some questions I had a few years ago that I never solved.
First, is there any application ...
- 3,107
12
votes
0
answers
373
views
Does each compact topological group admit a discontinuous homomorphism to a Polish group?
A compact topological group $G$ is called Van der Waerden if each homomorphism $h:G\to K$ to a compact topological group is continuous. By a classical result of Van der Waerden (1933) the groups $SO(...
- 41.9k
3
votes
1
answer
141
views
Measure on orbits of $N$ under conjugation by $H$
Let $G$ be a locally compact topological group with closed subgroups $H, N$ and $H$ normalizing $N$. Then $H$ acts continuously on $N$ by conjugation. If it will help, assume that $N$ is nilpotent, ...
- 11.6k
0
votes
0
answers
220
views
short exact sequence of profinite groups
Let $A\rightarrow B\rightarrow B/A$ be a short exact sequence of topological groups. Is it true that if there exists a continuous function $B/A\rightarrow B$ (of underlying spaces) such that the ...
- 1,613
4
votes
0
answers
156
views
Basic calculus on topological fields
Let $K$ be a a topological field (I am mainly interested in the cases when K is either an ordered field or a valued field, e.g. $K = \mathbb Q$ or $ \mathbb Q_p$).
1) Let $f: K^n \to K$ be a ...
5
votes
1
answer
198
views
A group with more than one Hausdorff minimal nontrivial group topologies
I have a few examples of a group $G$, equipped with a Hausdorff minimal nontrivial group topology $\cal T$. This means that $\cal T$ is Hausdorff and for any nontrivial (not necessarily Hausdorff) ...
- 1,145
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 ...
- 63.9k
8
votes
0
answers
570
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 $G$ ...
CommunityBot
- 1
31
votes
2
answers
3k
views
Is a normed space which is homeomorphic to a Banach space complete?
I have a normed space $(E,||\cdot||)$ which is homeomorphic (as a topological space) to a Banach space $F$.
Does this imply that $(E,||\cdot||)$ is also a Banach space?
I think I read something ...
- 4,713
4
votes
1
answer
152
views
Kind of multiplicative total boundedness in Hausdorff compact rings
Let $(R,\cal T)$ be a unital Hausdorff compact topological ring and let $A$ be an open subset of $R$ containing $1$. Is there a finite set $B$ with $AB=R$?
- 11.6k
6
votes
1
answer
249
views
Extending the topology on a set to the group/vector space it generates
The multiplicative group $\Bbb Q^+$ can be viewed as a $\Bbb Z$-module. To see this, note that any rational can be decomposed into the form
$2^{n_2} \cdot 3^{n_3} \cdot 5^{n_5} \cdot ...$
The tuple ...
- 4,907
0
votes
0
answers
85
views
Right split for homomorphism onto $S_\infty$
Let $G$ be a closed subgroup of $S_\infty$ and let $f:G\rightarrow S_\infty$ be a continuous surjective homomorphism. Under which conditions $f$ has a right split, i.e. there exists some $g:S_\infty\...
- 2,882
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 ...
- 5,409
3
votes
0
answers
182
views
LCH topologies on Groups that are not group topologies
Ellis's 1957 paper on Locally Compact Transformation groups proves the following:
A locally compact hausdorff topology on a group $(G, \cdot)$ for which left and right multiplication are (separately)...
- 352
7
votes
2
answers
472
views
A non locally compact group of finite topological dimension?
Is there a topological group which is Hausdorff, first countable,
locally connected and has finite topological dimension, yet fails
to be locally compact?
- 41.9k