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**15**

votes

**4**answers

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### 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 ...

**31**

votes

**7**answers

3k views

### Why are free groups residually finite?

Why is it that every nontrivial word in a free group (it's easy to reduce to the case of, say, two generators) has a nontrivial image in some finite group? Equivalently, why is the natural map from a ...

**28**

votes

**3**answers

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 ...

**17**

votes

**1**answer

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 ...

**7**

votes

**2**answers

423 views

### Potential connected non-Lie subgroup

This painful question is inspired by the question
"non-Lie subgroups" . Let R denote the real numbers. Let f be an discontinuous additive map from R --> R. Is it possible that the graph of f, inside ...

**7**

votes

**2**answers

517 views

### Avoiding countable subgroups of general uncountable groups

The following problem is a general form of another problem (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not ...

**14**

votes

**0**answers

499 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 ...

**13**

votes

**7**answers

2k views

### Haar measure on a quotient, References for.

I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it(thanks to some comments by Ben Linowitz).
Right from the very beginning, Weil uses the ...

**9**

votes

**2**answers

557 views

### Is there a non-trivial topological group structure of $\mathbb{Z}$?

More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?

**17**

votes

**4**answers

2k 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 continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. ...

**7**

votes

**1**answer

400 views

### Existence of infinite groups that are too reluctant to be topological

With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...

**5**

votes

**2**answers

294 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 ...

**4**

votes

**1**answer

128 views

### If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...

**3**

votes

**1**answer

404 views

### Understanding Bruhat's notion of Schwartz function

I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwartz-Bruhat space ...

**10**

votes

**3**answers

732 views

### How do you recover the structure of the upper half plane from its description as a coset space?

This is maybe a dumb question. $SL_2(\mathbb{R})$ has a natural action on the upper half plane $\mathbb{H}$ which is transitive with stabilizer isomorphic to $SO_2(\mathbb{R})$. For this reason, ...

**4**

votes

**2**answers

151 views

### Root of positive function in Fourier algebra

Let $G$ be a locally compact group, let $A(G)$ be the Fourier algebra of $G$. We think of $A(G)$ as a subalgebra of $C_0(G)$.
Question 1: Let $f\in A(G)$ be a function that is pointwise positive. ...

**4**

votes

**3**answers

568 views

### Action of a profinite group

Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod_{i \in I} \mathbb{Z}_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be ...

**3**

votes

**4**answers

253 views

### Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior?

Q1.
Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite ...

**2**

votes

**0**answers

20 views

### Can each non-open analytic subgroup of a Polish abelian group be covered by countably many closed Haar null subsets?

By a result of Laczkovich (http://www.ams.org/journals/proc/1998-126-06/S0002-9939-98-04241-5/S0002-9939-98-04241-5.pdf), any non-open analytic subgroup of a Polish locally compact group can be ...

**2**

votes

**0**answers

31 views

### Is the sumset of two Haar positive closed subsets of a Polish group non-meager?

A famous Steinhaus theorem says that if measurable subsets $A,B$ of a locally compact topological group $G$ have positive Haar measure, then the difference $AA^{-1}$ is a neighborhood of the unit and ...

**6**

votes

**2**answers

378 views

### How big $|Aut(M)|$ can be, given $|\partial Aut(M)|$?

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**

votes

**1**answer

174 views

### Are infinite groups “locally topologizable”?

Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point?
The question is inspired by and related to ...

**2**

votes

**1**answer

108 views

### Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods

Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?

**0**

votes

**1**answer

76 views

### Parallel group topologies on Prüfer groups

Let $p$ be a prome number. Are there group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Z_{p^\infty}$ such that
$$\mathcal T \nsubseteq \mathcal S,~~\mathcal S \nsubseteq \mathcal T$$