All Questions
Tagged with gn.general-topology topological-groups
213 questions
13
votes
1
answer
853
views
Mistake on article about Bohr compactification?
$\DeclareMathOperator\b{b}\newcommand\B{{\operatorname B}}$I wish to get help understanding the content of two theorems of [Iva] that seem mutually contradictory. First some context. Let $\b(\mathbb{R}...
- 11.4k
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^* \...
- 105k
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 $G$...
- 11.4k
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 ...
- 361
30
votes
2
answers
2k
views
Is every connected subgroup of a Euclidean space closed?
The question listed above (in the context of the complex numbers, but it is a reasonable question to ask in any dimension) was asked by a student in my complex analysis class, and I did not have an ...
- 41.9k
51
votes
5
answers
9k
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Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
- 35.5k
2
votes
1
answer
49
views
Is any submetrizable linear topology linearly submetrizable?
Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$.
Is ...
- 16.4k
3
votes
1
answer
529
views
Zeros of a function defined on $\mathbb{S}^2 \times \mathbb{S}^2$
Let $u$ be a smooth function on the sphere, and for each $y \in \mathbb{S}^2$, let $R_y$ be the $180^\circ$ rotation about the vector $y$. For each pair $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...
- 3,599
3
votes
0
answers
135
views
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=\{\...
- 3,011
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-...
- 12.9k
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
4
votes
1
answer
252
views
Does every (Abelian) Polish group have a nontrivial locally compact subgroup?
The question is pretty much in the title, suppose that $G$ is an (Abelian) nontrivial Polish group, must $G$ have a nontrivial locally compact (in the induced topology, hence necessarily closed) ...
- 655
2
votes
1
answer
213
views
Parametrization of topological algebraic objects
There are several results of the following form: if an algebraic objects is endowed with a topology (or rather uniformity) which is somehow compatible with the algebraic structure, this uniformity is ...
CommunityBot
- 1
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 ...
- 63.9k
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
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
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 ...
- 1
1
vote
0
answers
51
views
Discreteness of $D^{-1}D$ given that $D$ is uniformly discrete
Let $G$ be a topological group with unit element $e$.
We say that $D\subseteq G$ is discrete if for all $x\in D$ there is a unit-neighborhood $U\subseteq G$ such that $x^{-1}D\cap U=\{e\}$. We say ...
- 151
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 ...
- 4,713
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 $...
- 658
1
vote
0
answers
48
views
Neighborhoods of idempotents in topological inverse semigroups
In a topological group, for any neighborhood $U$ of the origin, there is another such neighborhood with the property that $V.V\subseteq U.$ I conjecture a similar property for topological inverse ...
- 63.9k
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
$\...
- 63.9k
0
votes
0
answers
67
views
G separable group, $\aleph_0 \leq \tau$. What is $l(X)$ and $\omega l(X) (\leq \tau)$? where $X \subseteq G$. And what is $\chi (G)$ (cardinal)?
Happy Chinese new year!
I was reading (and translating) a Russian article "On the topological groups close to being Lindelöf".
Where it is assumed G is a separable group and $\tau \geq \...
- 63.9k
16
votes
1
answer
481
views
Where can I learn more about the topology on $\mathbb{R}$ induced by the map $\mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$?
Consider the (continuous, injective, abelian group homomorphism) map $\Phi \colon \mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$ (where the target is given the product topology) taking $x\in \...
- 31.5k
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
12
votes
1
answer
624
views
Stone–Čech compactification as a semigroup
Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left ...
- 38.6k
1
vote
0
answers
81
views
Morphism in commutative square strict?
Let $G,H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism.
Then $f$ is said to be strict if $G/\mathrm{Ker}(f) \cong \mathrm{Im}(f)$ is an isomorphism of topological ...
- 473
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\...
- 41.9k
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 ...
- 21.3k
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 $\...
- 4,713
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
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
$$...
- 20.9k
3
votes
0
answers
124
views
Initial topology for a topological ring
Given a topological ring $R$ and an arbitrary (thus not necessarily surjective) epimorphism $q: R \to S$ of underlying rings is there a finest topology on $S$ such that 1) $S$ is a topological ring ...
- 103
1
vote
1
answer
190
views
Approximations by compact sub-spaces
Suppose $X$ is a Hausdorff (I'm happy to also assume "non compact") topological space that can be written as the topological direct limit
$$\varinjlim_{a\in J} K_a$$
for $J$ a directed set ...
- 12.9k
9
votes
1
answer
825
views
Is there a natural topology on the automorphism group of a topological group?
$\DeclareMathOperator\TAut{TAut}\DeclareMathOperator\Homeo{Homeo}$Let $G$ be a topological group, and let $\TAut(G)$ denote the group of topological automorphisms of $G$ under composition (i.e. the ...
- 12.9k
3
votes
3
answers
502
views
Is $(\mathbb{Z}_p\times \mathbb{R})/\mathbb{Z}$ connected?
I was reading this question The connected component of the idele class group but I am very confused about the structure of the solenoids $(\widehat{\mathbb{Z}}\times\mathbb{R})/\mathbb{Z}$, (where $\...
- 105k
10
votes
0
answers
272
views
What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?
What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$
I know that neither ...
- 3,011
8
votes
1
answer
442
views
Topological group locally homeomorphic to the Hilbert cube
Does there exist a topological group which is locally homeomorphic to the Hilbert cube $[0,1]^{\mathbb N}$?
Let me note that Hilbert cube has the fixed point property and thus it is not homeomorphic ...
- 63.9k
0
votes
0
answers
75
views
Is second countability an extension property for non-Hausdorff spaces?
Let $G$ be an abelian topological group and let $H$ be a non-Hausdorff closed subgroup (so that $G/H$ is Hausdorff). If $H$ and $G/H$ are second countable, is $G$ second countable?
5
votes
1
answer
155
views
Can the Boolean group $C_2^\omega$ be covered by less than $\mathfrak b$ nowhere dense subgroups?
Let $\mathrm{cov}_H(C_2^\omega)$ be the smallest cardinality of a cover of the Boolean group $C_2^\omega=(\mathbb Z/2\mathbb Z)^\omega$ by closed subgroups of infinite index. It can be shown that
$$\...
- 41.9k
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 ...
- 3,011
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.
...
- 11.4k
3
votes
1
answer
404
views
Picking a representative in a continuous way
I'm hoping for some ideas/pointers here. I'm experimenting with a Livschitz theorem for functions on a locally compact Abelian group, where the periodic orbit sums take values in a closed subgroup.
...
- 63.9k
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 $\...
- 9,486
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}...
- 63.9k
12
votes
0
answers
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
1
vote
1
answer
155
views
Do Locally Contractible, Path-Connected Groups have Accessible Bases?
Suppose $G$ is a locally contractible, metric, path-connected topological group. In my particular case, $G$ will be the group of orientation-preserving homeomorphisms of the plane, denoted $Aut(\...
- 439
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
4
votes
2
answers
634
views
Question about additive subgroups of the real line and the density topology
I am new studying additive subgroups of the real line, I would like to know if someone could give me an idea for the next question.
Let $m$ be the Lebesgue measure in $\mathbb{R}$. A measurable set $E\...
- 561
1
vote
1
answer
261
views
CH and the density topology on $\mathbb{R}$
In the article AN EXAMPLE INVOLVING BAIRE SPACES (https://www.ams.org/journals/proc/1975-048-01/S0002-9939-1975-0362249-1/S0002-9939-1975-0362249-1.pdf) of H. E. White Jr. it is shown that, assuming ...
- 11.4k