All Questions
Tagged with gn.general-topology topological-groups
71 questions with no upvoted or accepted answers
19
votes
0
answers
703
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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
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
12
votes
0
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349
views
Metric completion of an algebraically closed field is algebraically closed?
Let $F$ be a complete metric topological field. Suppose there is a subfield $F_1 \subset F$, algebraically closed and topoolgically dense in $F$. Must $F$ itself be algebraically closed?
We can ...
- 41.1k
12
votes
0
answers
172
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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
12
votes
0
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372
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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
11
votes
0
answers
273
views
A ZFC-example of a countably compact paratopological group which is not a topological group
Problem. Does there exist a ZFC-example of a countably compact Hausdorff paratopological group which is not a topological group?
(The problem posed 27 May 2015 by Alexander Ravsky on page 9 of Volume ...
- 4,535
10
votes
0
answers
272
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What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?
What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$
I know that neither ...
- 3,011
8
votes
0
answers
192
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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
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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
8
votes
0
answers
570
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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
0
answers
174
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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 ...
- 41.8k
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 ...
- 41.8k
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
5
votes
0
answers
269
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$T_1$ 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
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0
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204
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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 ...
- 41.8k
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
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0
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316
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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
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0
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128
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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 ...
- 41.8k
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
5
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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
votes
0
answers
138
views
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
5
votes
0
answers
164
views
Group topologies on $\Bbb Z$ with dense open sets in $\Bbb T$
Let $\Bbb Z$ be embedded in the circle group $\Bbb T$ by an irrational rotation and regard $\Bbb Z$ as a subgroup/subspace of the topological group $\Bbb T$.
Are there group topologies $\mathcal A$ ...
- 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
5
votes
0
answers
204
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Shrinking Group Actions
This is a repost from stackexchange. I didn't get an answer, so I figured I'd ask it here.
Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a ...
- 2,751
4
votes
0
answers
147
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Does the self-homeomorphism group of a finite CW complex have CW homotopy type?
Let $X$ be a finite CW complex and form the group $\mathcal{H}(X)$ of self-homeomorphisms $X\xrightarrow{\cong}X$, furnishing it with the compact-open topology. Under the assumptions on our space $\...
- 5,596
4
votes
0
answers
104
views
Does the group of homeomorphisms of the hilbert cube have automatic continuity
A topological group is said to have automatic continuity if every homomorphism from it to a second countable topological group is continuous. Various topological groups are known to have this ...
- 406
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
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 ...
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 ...
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
3
votes
0
answers
135
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What is the universal/fine uniformity on a topological group?
Cross posted from https://math.stackexchange.com/questions/4889335
I'm aware that every topological group is uniformizable: given a neighborhood $U\in\mathcal N(e)$ of the identity, the set $D_U=\{\...
- 1,322
3
votes
0
answers
31
views
Compactness of the minimal ideal of a compact Hausdorff polytopological semigroup
A semigroup $X$ endowed with a topology is called
$\bullet$ a topological semigroup if the semigroup operation $X\times X\to X$ is continuous;
$\bullet$ a semitopological semigroup if for every $a,b\...
- 41.8k
3
votes
0
answers
124
views
Initial topology for a topological ring
Given a topological ring $R$ and an arbitrary (thus not necessarily surjective) epimorphism $q: R \to S$ of underlying rings is there a finest topology on $S$ such that 1) $S$ is a topological ring ...
- 103
3
votes
0
answers
143
views
Does the compact-open topology retain topological groups?
Let $X$ be a topological space and $Y$ a topological group. Then $C(X,Y)$ is a group, and can also be endowed with the compact-open topology.
Is $C(X,Y)$ in the compact-open topology necessarily a ...
- 31
3
votes
0
answers
75
views
Are $T_0$ topological quasigroups completely regular?
In 1957 H. Salzmann generalized to quasigroups but weakened the standard result that $T_0$ topological groups are completely regular. He was able to show that $T_0$ topological quasigroups are regular ...
- 475
3
votes
0
answers
64
views
Metrically homogeneous spaces as inverse limits
Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following:
Is $X$ ...
- 541
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
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
190
views
What is the smallest number of nowhere dense affine subsets covering a topological group?
$\DeclareMathOperator\cov{cov}\newcommand\A{\text A}$A subset $A$ of a group $G$ is called affine if $A=xHy$ for some subgroup $H\subseteq G$ and some $x,y\in G$.
Given a non-discrete topological ...
- 41.8k
2
votes
0
answers
406
views
Complete topological groups in which all subgroups are closed
My previous question has been answered by YCor; so I am asking a new one with a reasonable additional assumption. See the previous question for the background and motivation.
General question: does ...
- 15.6k
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
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
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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
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 $...
- 41.8k
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
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
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
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
2
votes
0
answers
199
views
Finite topological dimension x local compactness
Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance:
A topological vector space is finite dimensional ...
- 4,729