All Questions
Tagged with gn.general-topology topological-groups
213 questions
5
votes
2
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454
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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
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 ...
- 41.8k
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 ...
- 41.8k
5
votes
0
answers
214
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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 ...
- 41.8k
7
votes
1
answer
207
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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 ...
- 4,413
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
2
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0
answers
49
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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 $...
- 41.8k
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
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 ...
- 41.8k
3
votes
0
answers
143
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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 ...
- 41.8k
1
vote
0
answers
206
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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 ...
- 41.8k
7
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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 ...
- 41.8k
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 ...
- 621
1
vote
0
answers
149
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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
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
8
votes
0
answers
306
views
Has the Roelcke completion of a topological group any reasonable algebraic structure?
It is well-known that each topological group $G$ carries (at least) four natural uniformities:
the left uniformity $\mathcal L$, generated by the base $\{\{(x,y)\in G\times G:y\in xU\}:U\in\mathcal ...
- 41.8k
12
votes
0
answers
372
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(...
- 41.8k
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
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
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, ...
- 6,180
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 ...
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
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 $G$ ...
- 4,588
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 ...
- 495
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
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 ...
- 4,897
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
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
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
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?
- 858
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
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 $AB=R$?
- 1,145
1
vote
0
answers
130
views
Not normal connected component of a right topological group
Let $\cal T$ be a locally compact topology on a group $G$ and $(x,y)\mapsto xy^{-1}$ be continuous at $(1,1)$ and for every $a\in G$, $x\mapsto xa$ be continuous everywhere with respect to $\cal T$.
...
- 1,145
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 ...
2
votes
1
answer
304
views
Can we Characterise Rings of Continuous Functions?
Suppose $K$ is some nice space, for example $\mathbb R$ or $\mathbb C$. Let $X$ be a set and $C$ a ring of functions $X \to K$. Is there any way to determine, from the algebraic structure of $C$, ...
- 1,955
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 ...
19
votes
0
answers
703
views
The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?
For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
- 3,104
3
votes
1
answer
142
views
What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite
When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism \begin{...
- 1,040
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
5
votes
0
answers
1k
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Examples of a topological semidirect product
Let $G$ be a compact topological group, and $\operatorname{Aut}(G)$ the group of autohomeomorphisms of $G$. I have proved some (topological) results about the holomorph $G\leftthreetimes \operatorname{...
- 203
5
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0
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138
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Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded?
Is there an example of an Abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...
- 1,145
14
votes
2
answers
1k
views
Baire Category Theorem for complete uniform spaces
The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...
- 1,270
0
votes
1
answer
178
views
Borel subsets of Polish groups
Suppose that I have a polish group $G$ and two subsets $A$ and $B$ of $G$ such that: $A$ is open in $G$ and $B$ is closed in $G,$ from this, can I conclude that $AB$ is a Borel subset of $G$? if not, ...
- 313
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