All Questions
Tagged with gn.general-topology topological-groups
213 questions
8
votes
1
answer
829
views
Topological groups in which all subgroups are closed
General question: does there exist a nondiscrete topological group $G$ such that all subgroups of $G$ are closed? Or, does there exist a nondiscrete topological vector space $V$ such that all vector ...
- 15.6k
8
votes
4
answers
3k
views
Finite dimensional vector spaces over a complete but not-necessarily-valued field
I'm essentially reopening this old question of Ricky Demer which was never fully answered.
Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and ...
- 2,585
8
votes
2
answers
362
views
Is every contractible homogeneous space of a connected Lie group homeomorphic to a Euclidean space?
Problem. Let $G$ be a connected Lie group and $H$ is a closed subgroup of $G$ such that the homogeneous space $G/H$ is contractible. Is $G/H$ homeomorphic to a Euclidean space $\mathbb R^n$ for some $...
- 41.9k
8
votes
1
answer
217
views
Why are free Boolean topological groups Hausdorff?
Assume $X$ is a Tychonoff space. Then $A(X)$ is the free topological abelian group over $X$. I know that $A(X)$ is Hausdorff and the canonical embedding from $X$ to $A(X)$ is a topological embedding.
...
- 83
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
8
votes
0
answers
192
views
Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?
I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question).
Some simple ...
- 60.5k
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 ...
- 41.9k
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 $G$ ...
- 4,588
7
votes
2
answers
653
views
The integers as a sequential but non-first countable topological group
Completely unaware of the Bohr topology, I recently asked whether or not there was a Hausdorff group topology on the integers $\mathbb{Z}$ which made the group fail to be first countable. For me, this ...
- 7,193
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
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 ...
- 4,413
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
7
votes
0
answers
174
views
Is each Choquet topological group strong Choquet?
A topological space $X$ is called (strong) Choquet if the player II has a winning strategy in the (strong) Choquet game.
It is known that a metrizable space $X$ is
$\bullet$ Choquet if and only if it ...
- 41.9k
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
7
votes
0
answers
433
views
Ever seen a ringed group?
A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...
- 12.3k
6
votes
3
answers
2k
views
Sequential topological vector spaces
Since I'm dealing with the distinction between sequential continuous and continuous maps at the moment I came to ask myself once again what can be said about spaces where these two notions agree (...
- 9,650
6
votes
2
answers
295
views
Embeds in a topological W-group, or a W-space that embeds in a topological group?
In Theorem 3.11 of Tkachuk - A compact space $K$ is Corson compact if and only if $C_p(K)$ has a dense lc-scattered subspace it's shown that if a compact Hausdorff space embeds in a topological W-...
- 1,352
6
votes
2
answers
507
views
Hausdorff group topologies on finitely generated groups
Suppose $G$ is a finitely generated Hausdorff topological group. Must $G$ be first countable (or perhaps a sequential space)? What if we restrict to the abelian case?
I wonder if this is even true ...
- 7,193
6
votes
2
answers
394
views
Intersection of all open subgroups vs. the intersection of all open normal subgroups
I am interested to know examples of topological groups $G$ for which the intersection $\bigcap\{H\leq G\mid H\text{ open}\}$ of all open subgroups of $G$ is the trivial subgroup but for which the ...
- 7,193
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 ...
- 61
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
6
votes
1
answer
727
views
Homomorphisms of Topological Groups which are Automatically Fiber Bundles?
Suppose I have a surjective homomorphism of topological groups $f:E \to G$. Let K be the kernel of f. The topological group K acts on E in an obvious way. When is this a fiber bundle over G? (It will ...
- 27.5k
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
6
votes
1
answer
191
views
Steinhaus number of a group
$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$.
Let $\mathcal A_X$ be the family of ...
- 41.9k
6
votes
2
answers
477
views
How big $|\operatorname{Aut}(M)|$ can be, given $|\partial\operatorname{Aut}(M)|$?
My apologies: There were a couple of typos in the original question. Hope I got them all.
Let $\kappa$ be an uncountable cardinal of cofinality $\omega$ and $M$ a model of size $\kappa$. We equip $\...
- 2,882
5
votes
3
answers
1k
views
On closed totally disconnected subgroups of connected real Lie groups
So the following statement seems to be obvious but I don't see how to prove it:
Q: How does one prove that a closed totally disconnected subgroup of a connected real Lie group is discrete?
Note that ...
- 7,609
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 ...
- 41.9k
5
votes
1
answer
877
views
Countable sum $\bigoplus_{n=0}^\infty\mathbb Z_p$ as a topological group
$\DeclareMathOperator\colim{colim}$This is inspired by Clausen's answer.
Question: Recall that $\mathbb Z_p$ is endowed with the $p$-adic topology. Consider the countable sum $M:=\bigoplus_{n=0}^\...
- 2,806
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
5
votes
1
answer
717
views
Structure of a profinite group as a condensed set with an action of an open subgroup
Let $G$ be a profinite group and $H$ be an open subgroup. As a continuous $H$-topological space, we have $G=\coprod_{G/H} H$. Does this also hold as condensed sets, i.e. do we have an identification ...
- 617
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
5
votes
2
answers
169
views
Local cross-sections for free actions of finite groups
Let $G$ be a finite group, let $X$ be a locally compact Hausdorff space, and let $G$ act freely on $X$. It is well-known that the canonical quotient map $\pi\colon X\to X/G$ onto the orbit space $X/G$ ...
- 717
5
votes
3
answers
676
views
Does every compact Hausdorff ring admit a decomposition into primitive idempotents?
Let $\mathbf{R} = (R,\mathcal{T},+,\cdot,0,1)$ be a compact Hausdorff topological unitary ring, and consider the set $I(\mathbf{R}) := \{ e \in R \mid e \cdot e = e \}$ (of idempotents in $\mathbf{R}$)...
- 1,498
5
votes
1
answer
247
views
How complex is the orbit equivalence relation of $\mathrm{Iso}_0(X)\curvearrowright S_X$ for $X=L^p([0,1])$?
For a Banach space $X$ let $S_X$ denote its unit sphere and let $\mathrm{Iso}_0(X)$ denote the group of rotations of $X$, that is isometries fixing the origin. There is a natural continuous action $\...
- 3,011
5
votes
2
answers
254
views
Empty interior of union of cosets?
The following question arises from trying to understand Lemma 1.3(ii) of arXiv:math/0405063. I believe a particular case of the proof (and in fact I think the proof is essentially equivalent to this ...
- 18.7k
5
votes
1
answer
101
views
continuity of certain map which is defined on a Stonean space
Let $G$ be a discrete group which acts continuously on a Stonean space $\Omega$. Consider the map $f\colon \Omega\to \{0,1\}^G$ sending $x\in \Omega$ to $\chi_{G_x}$, where $\chi_{G_x}$ denotes the ...
- 1,188
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
5
votes
1
answer
403
views
Is an extension of compact Hausdorff topological groups compact?
Let $1 \rightarrow A \xrightarrow{a} B \xrightarrow{c} C \rightarrow 1$ be a short exact sequence of topological groups (i.e., all maps are continuous, $A = \mathrm{Ker}(c)$, and $C = \mathrm{Coker}(a)...
- 1,103
5
votes
1
answer
186
views
When does topological homogeneity imply algebraic homogeneity? Pseudo-arc and Hilbert cube
Knaster's pseudo-arc and Hilbert cube are topologically homogeneous continua. The easier question is: do these spaces admit a topological group structure? (I am sure that the answer is negative). Thus ...
- 7,500
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
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
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\...
- 41.9k
5
votes
1
answer
155
views
Can the Boolean group $C_2^\omega$ be covered by less than $\mathfrak b$ nowhere dense subgroups?
Let $\mathrm{cov}_H(C_2^\omega)$ be the smallest cardinality of a cover of the Boolean group $C_2^\omega=(\mathbb Z/2\mathbb Z)^\omega$ by closed subgroups of infinite index. It can be shown that
$$\...
- 41.9k
5
votes
1
answer
329
views
Example of a quasitopological group with discontinuous power map
A quasitopological group is a group $G$ with topology such that multiplication $G\times G\rightarrow G$ is continuous in each variable (i.e. all translations are continuous) and inversion $G\...
- 7,193
5
votes
0
answers
269
views
$T_1$ paratopological group having a dense commutative subgroup is commutative
I'm learning about topological groups from Arhangelskii and Tkachenko "Topological groups and related structures" and this is one of the exercises there.
A paratopological group is a group ...
- 1,201
5
votes
0
answers
204
views
What are all of the topological (commutative) monoid structures on a closed interval?
Consider a closed real interval $[a,b]$ as a toplogical space. Up to homeomeorphism it doesn't matter, but I like to take $[a,b] = [0,\infty]$.
Question 1: What are all of the topological commutative ...
- 63.9k
5
votes
0
answers
143
views
Two cardinal characteristics of the continuum, related to the Bohr topology on integers
For a subset $A\subseteq\mathbb T$ of the unit circle $\mathbb T=\{z\in\mathbb C:|z|=1\}$, let $\tau_A$ be the smallest topology on the additive group of integers $\mathbb Z$ such that for every $z\in ...
- 41.9k
5
votes
0
answers
129
views
Is there an orbit map without path lifting property?
I am looking for an example of a topological group $G$ acting by homeomorphisms on a metrizable space $X$ such that the orbit map $X\to X/G$ doesn't have the path lifting property, that is, there is a ...
- 29.1k
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
0
answers
128
views
Is the Baireness a 3-space property of topological groups
It is known that the product of two Baire spaces can be meager.
On the other hand, by a recent result of Li and Zsilinszky the product of two Baire spaces is Baire if one of the spaces is countably ...
- 41.9k