The topological-groups tag has no wiki summary.

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### 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 ...

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### A group allowing exactly 7 group topologies

Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{trivial}, \mathcal T_{discrete}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with
$$\mathcal ...

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### 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
$$\mathcal T \nsubseteq \mathcal S,~~\mathcal S \nsubseteq \mathcal T$$

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### Equations and random subgroups in compact groups

EDIT: Here is a more specific question.
Let $G$ be a compact group and let $w$ be a word in $d$ variables. Then the solution set $S$ of the equation of $w=1$ is a closed subset of the product $G^d$ ...

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### 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 ...

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### 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 $h \in U_h$.
...

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### Infimum of two group topologies

Let $G$ be a non-abelian group and $\mathcal S$ and $\mathcal T$ be group topologies on $G$. What is the largest group topology $\tau$ on $G$ with $\tau \subseteq \mathcal T\cap \mathcal S$?
In ...

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### 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 = ...

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### Union of conjugates of a closed subgroup of a compact group

Let $G$ be a compact Hausdorff group, $H \leq G$ a closed subgroup of infinite index in $G$.
Is it possible that the conjugates of $H$ cover some open neighbourhood of $1$ in $G$ (or the whole of ...

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### The set of (property) elements of a locally compact group is closed

For which properties $(P)$ is the following statement known to be true?
In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the ...

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### Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)

Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
...

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### Topological proof of a result in Logic

I proved the result below using logic. My questions:
Can this theorem be proved by purely topological means?
Do you know any theorems that either can be used to prove the same result, or which give ...

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### Simple but topologizable [closed]

Do you have an example of an infinite simple group with at least 3 distinct group topologies on it?

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### Locally compact vs. compactly generated in group theory

I'm studying the Creutz-Shalom paper on the normal subgroup theorem for dense commensurators of lattices. The authors state their results for locally compact and compactly generated groups. I do not ...

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### 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 $G\to G,~~ a\mapsto a^{-1}$ ...

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### The group of diffeomorphisms with compact support

Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms with compact support, turning it into a (locally-)compact topological group?
(My ...

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### In what topological abelian groups does convergence to zero imply summability?

(This question has been on math.SE for over a week and has not gotten any answers.)
Let $G$ be a (T$_0$) topological abelian group, and let $0$ be its identity element.
Assume that for all index ...

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### Compact Quantum Groups and the Existence of the Classical Haar Measure

Before I state my question, let me provide the definition of a compact quantum group.
Definition: An ordered pair $ \mathscr{G} = (\mathscr{A},\Phi) $ is called a compact quantum group if
...

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### Haar measurable sets and quotient maps

Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...

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### Free profinite completions

Let $m,n \in \mathbb{N}$. Which residually finite groups $G$ generated by $m$ elements, have the free profinite group on $n$ generators as their profinite completion?

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### Automorphisms of profinite groups

Let $d,n \in \mathbb{N}$, and $p$ a prime number. Let $F$ be a free pro-$p$ group on $d$ generators. Is there an automorphism of $F$ of order $n$?

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### 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 ...

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### Can an acyclic continuum be metrically homogenous? (I'd say: no way! :-)

I asked recently on MO about algebraic structures admitted by topologically homogenous continua like the Hilbert cube $\ I^{\mathbb N}\ $ or the Knaster pseudo-arc. There is a relation between the ...

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### 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 ...

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### rationalization of classifying spaces

This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway:
Let $G$ be a simply-connected topological group. In particular, it is an ...

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### 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 ...

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### Automorphism group of compact abelian group

I am looking for references on the automorphism group $\mathrm{Aut}(X)$ of a compact abelian group $X$. By automorphisms I mean topological group automorphisms. Some particular questions are as ...

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### Hausdorffness inheritance in topological groups

Suppose $\mathcal T$ and $\mathcal S$ are two compatible Hausdorff topologies on a group $G$ and $\mathcal R$ is a maximum compatible topology on $G$ with $\mathcal R \subseteq \mathcal T\cap \mathcal ...

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### Semi-continuity of injectivity radius

Let $H$ be a locally compact second countable topological group and $X$ be a locally compact second countable space. Assume that $H$ acts continuously on $X$ and that for every $x$, $H_x=\{h \in H ; ...

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### Connectedness properties of groups of homeomorphisms

Denote by $H(X)$ the group of homeomorphisms of a topological space $X$. Assume further that $X$ is either compact or locally compact and locally connected. In both cases $H(X)$ becomes a topological ...

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### injective implies completion injective?

Background and definitions.
Let $k$ denote a field complete with respect to a non-trivial non-archimedean norm. Let $R$ be the integers in $k$, and say $\pi\in R$ with $0<|\pi|<1$ ($\pi$ ...

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### Locally finite compact groups

I assume all tolpological groups here to be Hausdorff. A group is called locally finite if every finitely generated subgroup is finite. What can be said about a locally finite compact group? Must it ...

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### Is a weakly separable group always Lindelöf?

By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...

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### Restriction of “$\pi_{1}$” to topological groups

Let $G$ and $H$ be two topological groups. Assume that $\phi:\pi_{1}(G) \to \pi_{1}(H)$ is a group homomorphism. Is there a continuous function $f:G\to H$ such that $f_{*}=\phi$?

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### Is a left topological group which is a manifold a topological group?

Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not ...

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### Continuity of inversion and composition in certain topological groups

For $k\in\Bbb{N}$ ($k\geqslant 1$) and $\alpha\in]0,1]$, let $H_{k,\alpha}([0,1])$ be the group of orientation preserving $C^{k,\alpha}$ diffeomorphisms of the closed unit interval $[0,1]$. We furnish ...

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### Is any continuous group homomorphism from R to C* an exponential map? [closed]

Consider $\mathbb{R}$ to be an additive topological group, and $\mathbb{C}^{\ast}$ to be a multiplicative topological group.
Is the following statement true?
If so, then how can one prove it?
...

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### Irreducible representations of compact groups

Let G be a compact group (or even profinite - Galois group). Let $V$ be a vector space over the field ${\mathbb F}_p$ with $p$ elements, $p$ a finite prime, such that $V$ is a contable product of ...

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### subsets of groups which have to be closed no matter what

One example of a subset of a group $G$ which has to be closed in any topology on $G$ compatible with the group operations is a centraliser. Are there any other interesting examples?

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### 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 Schwarz-Bruhat space ...

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### Group morphism and axiom of choice

Let $n$ be a strictly positive natural integer. Let us consider the topological group $(\mathbb{R}^n,+)$ with its usual structure.
In ZF, can we deduce some form of the axiom of choice from the ...

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### Do syndetic sets on amenable semigroups have positive upper density?

Let $\mathbb{G}$ be a discrete amenable semigroup, and $\left\{ F_{n}\right\} $
a Folner sequence.
For $S\subset \mathbb{G}$ define the upper density as $D^{\ast
}(S)=\limsup_{n\rightarrow \infty ...

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### Tensor product of topological abelian groups with the reals

Given an abelian group A, the tensor product $A \otimes R$, where R are the reals, is naturally an R-vector space.
Now suppose that A is a topological abelian group (if necessary, we can assume it ...

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### Is the kernel of the Bohr compactification minimally almost periodic provided that it is cocompact?

Let $G$ be a locally compact (second countable) group and let
$$
G_0 = \cap \{ \ker\pi : \pi \text{ is a continuous finite-dimensional unitary representation of } G \}.
$$
This is the kernel of the ...

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### The finest countably generated free topological group so that $x^{m_{n}}_n\rightarrow 1$?

What is the finest free topological group $H$ with generators ${x_{1},x_{2},...}$ so that $x^{m_{n}}_n\rightarrow 1$ for all sequences $m_{1},m_{2},...$?
Is $H \simeq K$, with $K$ the natural ...

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### Existence of $n$-connected topological groups with $m$-dimensional action extending that of $GL(m)$ on $\mathbb{R}^m$

I'll first state the question as concisely as I can and then provide some motivation.
Consider two positive integers $m$ and $n$ such that $m < n+2$. Does there exist a topological group $G$ ...

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### Completion of abelian topological groups

During some exercises I came upon the following questions that I cannot answer myself. Given a topological group $G$ we can build the completion using cauchy sequences and denote this completion by ...

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### topological group that is connected and locally connected but not path-connected

Is there a ($\mathrm{T}_0$) topological group that is connected and locally connected but is not path-connected?
This is a cross-post from MSE, since my question there was posted over three weeks ...

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### Is every regular paratopological group completely regular?

This problem is presented as an open problem 1.31. on p.26 of Arhangel'skii-Tkachenko, Topological groups and related structures. Is this problem still open?
Dusan

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### To what extent has the Haar measure been generalized?

It is known that all locally compact groups, and therefore compact groups, have a left-invariant Haar measure which is unique up to scalar constant, also a right-invariant one. Is there a strictly ...