# Tagged Questions

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### What does it mean for a group to act cocompactly by isometries on a topological space $X$?

What does it mean for a group to act cocompactly by isometries on a topological space $X$? I know if $X$ is a topological group, and $A$ a subspace, then $A$ is cocompact iff $X/A$ is compact. Not ...
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### Proving closure of unit space of a Hausdorff groupoid

For Hausdorff topological groups, the set $\{e\}$ containing only the identity is closed. This is because Hausdorff implies T1 which implies singletons are closed. For topological groupoids, defined ...
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### Profinite topology of a Group

Let $G$ be a group. Consider now the set of all (left for instance) cosets in $G$ of subgroups of finite index. This set is a base for a topology in $G$. I found somewhere that if $G$ is residually ...
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### Explanation of a passage in Atiyah / Macdonald

On page 105 the authors show that $\hat{\hat{G}} \cong \hat{G}$ (Proposition 10.5) and conclude that the canonical homomorphism $\phi : \hat{G} \to \hat{\hat{G}}$ is an isomorphism. How does the fact ...
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### A homeomorphism of topological groups.

Let $L/K$ be a Galois field extension and $(I, \leq)$ be the directed set of all finite Galois extensions $E$ of $K$ contained in $L$ (we say $E^{\prime}\leq E$ if $E^{\prime}\subseteq E$). If ...
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### Continuous maps of topological groups

Suppose $\theta: G\to K$ is a homomorphism of topological groups. We know that $G$ and $K$ each have fundamental systems of open neighbourhoods of the identity. If I want to show that $\theta$ is ...
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### Closure of a topological group

Let $G$ be a topological group. Define $H(g)=\{g^n\}^{\infty}_{n=-\infty}$ for each $g \in G$. I need to prove that the closure of the set $H(g)$ is a commutative subgroup of $G$. But I am not sure ...
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### center of the group of orthogonal matrices of dim 3 [duplicate]

i am looking for the center of the group of orthogonal matrices of dimension 3. i'm thinking it contains all rotations and reflections but i'm not sure i'm correct and (assuming i am) don't know how ...
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### Is true that $Z(G)/N = Z(G/N)$ for connected topological groups?

Let $G$ be a connected topological group and $N$ a discrete normal subgroup of $G$. Is it true that $Z(G)/N = Z(G/N)$, where $Z(G)$ denotes the center of $G$? I know that every discrete normal ...
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### Connected subgroups of SU(2) and SU(3)

I am reading 'Lie groups, Lie Algebras, and Representations : An Introduction' by Brian Hall and am unable to do the problem 17 in chapter 3. It says Show that every connected Lie subgroup of ...
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### How many compatible group structures does a topological space admit?

Suppose I have a topological space $(G,\tau)$ and am interested in whether there exists a topological group $(G,*,\tau)$. In other words, can we assign a binary operation $*$ to $(G,\tau)$ which ...
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### Please list a few topological groups that I should learn about.

I'm going through Munkres' Topology book and there's a lot about topological groups. For fear that I'll forget the theorems on them I'd like to connect each thing I prove with a real-world example. ...
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### A characterization of the module function on a locally compact division ring

References: Weil's Basic Number Theory(written as BNT). Bourbaki's Commutative Algebra(written as BCA). Let $K$ be a topological ring with an identity. Suppose every non-zero element of $K$ is ...
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### $G$ is locally compact semitopological group. There exists a neighborhood $U$ of $1$ such that $\overline{UU}$ is compact.

Let $G$ be a group and $\mathcal T$ be a locally compact topology on $G$ such that for any $a\in G$ the functions $x \mapsto ax$ and $x\mapsto xa$ are continuous on $G$. How to prove elementarily ...
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### An example of a compact multiplicatively unbounded ring

My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
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### Cauchy product on topological rings

Let $R$ be any commutative Hausdorff topological ring. I am looking for a preferably general condition on sequences $(x_n)_{n \in \mathbb{N}}$, $(y_n)_{n \in \mathbb{N}}$ such that the equation  ...
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### The intersection of open normal subgroups in a compact, totally disconnected topological group is trivial.

I am currently doing self-study on profinite groups and I'm stuck trying to prove the following lemma. If a topological group $G$ is compact and totally disconnected, then the open normal ...
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### Normal subgroups of the Special Linear Group

What is some normal subgroups of SL(2, R)? I tried to check SO(2, R), UT(2, R), linear algebraic group and some scalar and diagonal matrices, but still couldn't come up with any. So can anyone give ...
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### Research Sources for $SL(2,R)$

Can anyone guide me to a good site for the special linear group $SL(2,R)$, especially one that goes deep into its subgroup and normal subgroup? Book recommendations would be great too.
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### Induced topology on the homomorphic image of a topological group

I would like to do a small sanity check on the following situation: Let $\pi: G \rightarrow G'$ be a surjective homomorphism of topological groups. Let the topology of $G$ be given by a sequence of ...
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### Topology induced by the completion of a topological group

Let $G$ be an abelian topological group and let $\hat{G}$ be its completion, i.e. the group containing the equivalence classes of all Cauchy sequences of $G$. What exactly is the topology of ...
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### Sum of Cauchy sequences is Cauchy in an Abelian Topological Group

Let $G$ be a topological abelian group and suppose $0$ has a countable fundamental system of neighborhoods. Let $(x_n),(y_n)$ be Cauchy sequences of $G$. Why is it true that $(x_n+y_n)$ is a Cauchy ...
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### Two Lie groups which are isomorphic but not homeomorphic

I am looking for an example of two Lie groups which are isomorphic as groups but not homeomorphic as topological spaces. Or, even more interestingly, a proof that two such groups cannot exist. Does ...
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### a question on normal subgroup of $GL_n(\mathbb{C})$ and $GL_n(\mathbb{R})$

I am really sorry that I am not able to solve this one, thank you for your help.
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### Definition of a topological module

A topological universal algebra of type $\Omega$ is a universal algebra $A$ of type $\Omega$ that is also a topological space, such that for any $n\!\in\!\mathbb{N}$ and any operation ...
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### Why metrizable group requires continuity of inverse?

A metrizable group is a metric space $(G,d)$ with a binary operation $\cdot$ such $(G,(\cdot))$ is a group and maps $(\cdot):G\times G\to G$ and $f:G\to G$ given by $(\cdot)(x,y)=xy$ and $f(x)=x^{-1}$ ...
Let $G$ be a topological group and $(X,\mu)$ be a $G$-set, i.e. $\mu$ defines an action $X \times G \rightarrow X$. Is it then true that $\mu$ is continuous if and only if for every $x \in X$ the ...
suppose $X$ is a topological space and $G$ and $H$ are groups acting on it. 1) if $G$ is isomorphic to $H$ do we have necessarely $X/G$ is homeomorphic to $X/H$ 2) suppose $G$ and $H$ are two ...