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12
votes
1answer
959 views

(Closures of sets of) operations in topological groups.

Let $G$ be a topological group. For each $n \in \mathbb{Z}$, consider the continuous functions $f_{n} \colon G \to G : x \mapsto x^{n}$, and set $F := \{f_{n} \mid n \in \mathbb{Z}\}$. Is there a ...
0
votes
1answer
567 views

No non-trivial homomorphism to a group

Here is a question I posted some months ago in Math.SE, and t.b. mentioned to the following question by Florent MARTIN which is somehow related to my question; Let $G$ be a compact Hausdorff ...
0
votes
0answers
110 views

A continuous map from a T2 & compact space to a uniform space is uniformly continuous.

Can you recommend some literature that give a proof of this statement, and who allegedly prove it first? BTW, is there any use of uniform spaces or topological spaces in mathematical (or theoretical) ...
26
votes
3answers
2k views

morphism from a compact group to Z ?

I wonder if it there exists a topological compact group $G$ (by compact, I mean Hausdorff and quasi-compact) and a non-zero group morphism $\phi : G \to \mathbb{Z}$ (without assuming any topological ...
3
votes
0answers
209 views

When Aut(M) preserves a linear order?

I have a general-type question: Suppose $M$ is a countable structure that is ultrahomogeneous, i.e. every (partial) isomorphism between finitely generated substructures of $M$ extends to an ...
0
votes
0answers
258 views

classifying continuous functions on the $ax+b$ subgroup of $GL(2,\mathbb R)$

Let $G$ be 'ax+b' topological group i.e subgroup of $GL(2,\mathbb R)$ containing $2\times 2 $ matrices of type $\{\left(\begin{matrix} a&b\\\0&1\end{matrix}\right): a\neq 0\; a,b\in \mathbb ...
2
votes
1answer
131 views

Categories with canonical factorizations into products satisfying two particular properties

An old splitting theorem for (Hausdorff) locally compact abelian (LCA) groups says that any LCA group $L$ is isomorphic to a direct product of $\mathbb{R}^n$ and $L_1$, where $L_1$ contains a ...
4
votes
1answer
229 views

Does the weak approximation theorem hold for general topological fields?

The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topology from $|\cdot|_i$, ...
16
votes
1answer
718 views

Does ZF prove that topological groups are completely regular?

Let $\mathbf{G} = \langle G,\cdot,\mathcal{T}\;\rangle$ be a topological group. Let $\mathbf{e}$ be the identity element of $\langle G,\cdot \rangle$. Assume $\{\mathbf{e}\}$ is closed in $\langle ...
3
votes
1answer
453 views

Complete metrics in locally compact topological groups

Hello, I am trying to show that every metrizable locally compact topological group admits a complete metric generating the topology of the group
0
votes
2answers
328 views

Basic question on minimal flows

I know that minimal flows are actions for which no proper closed invariant subsets exist, but I am unclear how to understand this concept. If a coset flow on a quotient space Gamma/S is ergodic, ...
6
votes
1answer
239 views

A restatement, in terms of the semi-group product of the left-invariant completion of a Polish group, of http://mathoverflow.net/questions/71389

This is a re-statement, of sorts, of Is there a relational countable ultra-homogeneous structure whose countable substructures do not have the amalgamation property?, so far unanswered. Let $G$ be a ...
7
votes
4answers
1k 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 ...
3
votes
2answers
269 views

When does a LCA group not contain a (closed) infinite cyclic subgroup?

If $G$ is an LCA (locally compact abelian) group, is there any 'nice' sufficient (or preferably necessary and sufficient) criteria for when $G$ does not contain a closed (and hence discrete in the ...
6
votes
1answer
959 views

Haar measure for large locally compact groups

In this answer, Gerald Edgar mentions that Haar measure is naturally defined on the $\sigma$-algebra of Baire sets (the smallest $\sigma$-algebra that contains all the compact $G_\delta$ sets) of a ...
6
votes
2answers
407 views

Topology on extensions of topological groups

Let $G$ and $H$ be two topological groups and let $\mathcal{E}:0 \to G \to E \to H \to 0$ be an extension of abstract groups. Is there a way to introduce a topology on $E$ such that $\mathcal{E}$ ...
1
vote
2answers
352 views

Lie (and topological) group extensions of $\mathbb{R}^2$ by $\mathbb{R}$

What are all the non-split Lie (and topological) group extensions $0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0$? Here, $\mathbb{R}$ and $\mathbb{R}^2$ are regarded as Lie (and topological) groups ...
5
votes
0answers
169 views

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
votes
1answer
491 views

On homeomorphic compact connected topological groups

I wish to thank Professor Claudio Gorodski for his very helpful answers to my question on the webcite: If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie ...
1
vote
1answer
279 views

Have a general mean of an amenable group a kind of Fubini's property ?

Recently in my investigations I faced a problem related to amenable groups. I have no idea if my question is suitable for this site, but after ask some collegues in my department I decided to try to ...
1
vote
0answers
169 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 ...
1
vote
0answers
194 views

exotic compact group

Let $G$ be compact (and Hausdorff) group, $\mu$ be Haar measure on $G$. Is it always true that $(G,\mu)$ is a standard probability space (Lebesgue-Rokhlin space)? It is so if (a priori not iff) the ...
14
votes
0answers
929 views

What groups are Lie groups?

We know how to tell if a topological group is a Lie group: this was famously asked by Hilbert and answered gloriously by Gleason, Montgomery and Zippin in the 50s (a locally compact topological group ...
6
votes
4answers
756 views

Criteria for topologically finitely generated profinite groups

Q1: Do we have a criterion which allows us to say when is a profinite group $G$ topologically finitely generated? For example, if $G$ is topologically finitely generated then, for a fixed integer ...
0
votes
3answers
640 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 ...
5
votes
3answers
668 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 ...
3
votes
1answer
311 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. ...
20
votes
0answers
900 views

Do all possible trees arise as orbit trees of some permutation groups?

I.Motivation from descriptive set theory (Contains some quotes from Maciej Malicki's paper.) The classical theorem of Birkhoff-Kakutani implies that every metrizable topological group G admits a ...
5
votes
2answers
654 views

Examples of totally disconnected, locally compact non-sigma-compact groups

I am looking for examples of totally disconnected, locally compact groups, which are not sigma-compact. For a start any such an example would do, so that I can a feeling for those groups and how to ...
15
votes
4answers
1k views

Compact open topology on $\mathrm{Homeo}(X)$

Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of contiunous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. ...
15
votes
1answer
1k views

Which principlal bundles are locally trivial?

If $H$ is a closed subgroup of a topological group $G$, then the orbit map $G\to G/H$ is a principal bundle, yet somewhat surprisingly, it need not be locally trivial. In the wikipedia article on ...
10
votes
4answers
906 views

When do isometric actions exist?

Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...
10
votes
1answer
430 views

Epimorphisms have dense range in TopHausGrp?

Consider the category of Topological Groups with continuous homomorphisms. Then a continuous homomorphism $f:G\rightarrow H$ with dense range is an epimorphism. Is the converse true? If not, what ...
11
votes
1answer
709 views

Non-isomorphic two-transitive permutation groups with isomorphic point stabilizers

The permutation groups $A = PSL(2,7)$ with its natural action on the projective line $\mathbb{P}^1(\mathbb{F}_7)$ and $B = A\Gamma L(1,8)$ with its natural action on the affine line $\mathbb{F}_8$ ...
9
votes
3answers
990 views

Why is the dual of a torus the same as its fundamental group?

The set of continuous homomorphisms from a torus ${\mathbb T}^n = ({\mathbb R}/{\mathbb Z})^n \to {\mathbb R}/{\mathbb Z}$ can be identified with ${\mathbb Z}^n$ if we assign to each $k = (k_1, \ldots ...
7
votes
2answers
1k views

Locally compact abelian groups

First, some preliminaries: Define an "LCA group" to be a locally compact Hausdorff abelian topological group. Define "smooth manifold" in a way that requires Hausdorffness, but not connectedness or ...
3
votes
1answer
363 views

Topological Groups and Families of Pseudometrics

The topology on a topological group is generated by a family of pseudometrics. The only proof I know passes through uniform spaces (by which I mean the entourage definition): A topological group has ...
2
votes
2answers
1k views

Irreducible unitary representations of locally compact groups

Let $G$ be a locally compact group and let $\mu$ be a left Haar measure. We know that $\mu$ is unique up to a scalar in $\mathbf{R}_{>0}$. I don't know so much about unitary representations of ...
12
votes
1answer
730 views

Is every compact topological ring a profinite ring?

There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...
1
vote
2answers
730 views

When is the group of homeomorphisms of a compact space locally compact?

When is the group of homeomorphisms of a compact space locally compact? I am interested in finding out when the group of homeomorphisms of a compact topological space $X$ (with appropriate ...
8
votes
1answer
519 views

index of a closed subgroup of a profinite group

In the book "profinite groups, arithmetic, and geometry" of Shatz, the index $(G:H)$ of a closed subgroup $H$ of a profinite group $G$ is defined to be the supernatural number $lcm\big((G/U):(H/(H\cap ...
-1
votes
1answer
847 views

Compact Topological Group Properties [closed]

I feel I want to understand it better. I know that for every cover there's a finite subcover but what can you say about it's group properties? I'm stuck on this homework problem where we were asked: ...
8
votes
0answers
408 views

Are there non-reflexive abelian topological groups isomorphic to their second dual?

I posted the following question in a comment at Are there non-reflexive vector spaces isomorphic to their bi-dual? and it got one upvote, but it didn't get an answer, so I'll post it as an ...
14
votes
0answers
485 views

Closed connected additive subgroups of the Hilbert space

It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is ...
5
votes
4answers
517 views

orbits in locally compact group

As everyone knows if $x\in S^1$, then the set $\{ x^n \}$ is either finite or dense. Under which condition is true for any other locally compact group, i.e if $G$ is a locally compact group, and $x\in ...
36
votes
6answers
3k views

Does homeomorphic and isomorphic always imply homeomorphically isomorphic?

Let $(G,\cdot,T)$ and $(H,\star,S)$ be topological groups such that $(G,T)$ is homeomorphic to $(H,S)$ and $(G,\cdot)$ is isomorphic to $(H,\star)$. Does it follow that $(G,\cdot,T)$ and ...
-1
votes
2answers
658 views

Is there a group homeomorphic to but not homeomorphically isomorphic to the circle group?

Let $Circ$ be the topological group $(\{z\in \mathbb{C} : \overline{z}\cdot z = 1\},\cdot , \{U\in 2^{\{z\in \mathbb{C} : \; \overline{z}\cdot z \, = \, 1\}} : \{z\in \mathbb{C} : \overline{z}\cdot z ...
15
votes
4answers
2k views

Fundamental groups of topological groups.

Let $G$ be a topological group, and $\pi_1(G,e)$ its fundamental group at the identity. If $G$ is the trivial group then $G \cong \pi_1(G,e)$ as abstract groups. My question is: If $G$ is a ...
7
votes
1answer
645 views

Which groups can be recovered from their unitary dual?

Note: in this post, every topological group under consideration is assumed to be Hausdorff. Given a locally compact abelian group, one can construct its dual group, i.e. its group of (unitary) ...
4
votes
1answer
206 views

existence of charaterization of amenable groups by complementation?

Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$. Do you know a charaterization of discrete amenable ...