# Tagged Questions

A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).

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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### A possible mistake in Walter Rudin, “Fourier analysis on groups”

I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$): Suppose $E$ is a coset in $\Gamma_2$ ...
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### 'Infinitesimal' elements of a topological group

Let $G$ be a topological group, and let $M$ be the intersection of all conjugacy-invariant neighbourhoods of the identity in $G$ (in other words, the set of elements that can be taken arbitarily close ...
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### 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 ...
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### 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 ...
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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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### Almost conjugation-invariant neighborhoods of units in locally compact groups

Let $G$ be a locally compact topological group with unity $e$ and left Haar measure $m$. Let also $g\in G$ be a given element and $U$ a neighborhood (of compact closure) of $e$. I am interested to ...
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### 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 ...
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### Group actions on principal groupoids

Suppose that $\mathcal{G}$ is etale principal groupoid and that $G$ is a discrete (or finite) group acting freely on the locally compact unit space $\mathcal{G}^0$ (or assuming compactness, if ...
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### Subgroups of finite non-zero Haar measure of abelian locally compact groups

Is it true that every subgroup of finite non-zero Haar measure of an abelian locally compact group should be open and compact? This is obviously true for the case of discrete abelian groups. Thanks.
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### Not normal connected component of a right topological group

Let $\cal T$ be a locally compact topology on a group $G$ and $(x,y)\mapsto xy^{-1}$ be continuous at $(1,1)$ and for every $a\in G$, $x\mapsto xa$ be continuous everywhere with respect to $\cal T$. ...
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### In a topological group $G$ with its lower uniformity, if $G$ is locally totally bounded, is its completion locally compact?

There has been work done on groups whose lower uniformity (or Roelcke uniformity) is totally bounded, e.g. the orthogonal group on a Hilbert space. This condition is equivalent to saying the lower ...
Is there any description known of the Hausdorff topologies on $\mathbb{Q}$ compatible with the group operations?
If $H$ is a strongly complete profinite group and $K$ is a dense countably-infinite-index subgroup, then I'm assuming a proper finite-index subgroup of $K$ could still be dense in $H$. Is there any ...