# Tagged Questions

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### What is a Topological Group Intuitively?

What is a topological group intuitively, beyond just being able to say things are near to each other in a group, and why is it a good idea to consider this theory as part of general topology as ...
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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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### Suppose that $X$ is locally compact and $G$ acting on $X$ is proper. Show that the quotient $X/G$ is Hausdorff.

Suppose that $X$ is locally compact and $G$ acting on $X$ is proper. Show that the quotient $X/G$ is Hausdorff. I am working through some notes on Geometric Group Theory and I am having a hard time ...
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### True or False: Topological Group and $S^1 \vee S^1$

$i.$ $S^1 \vee S^1$ can be embedded in a topological group $ii.$ $S^1 \vee S^1$ can be covered by a topological group I think $i.$ is true since we can embed the wedge sum into $\mathbb{R}^2$, which ...
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### Creating a Topological group from modulo multiplication Group.

If I were to create a Topology out of the Modulo 3 Multiplication group $\mathbb{Z}_3$, what elements would it consist of and why? So $\mathbb{Z}_3 = \{0,1,2\}$ as a group over modulo 3. What are the ...
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### Closed and Connected subgroups of $\mathbb{R}^n$

Question is : What are closed connected subgroups of $\mathbb{R}$ and from that deduce what are closed connected subgroups of $\mathbb{R}^n$ What i have done so far is : Only connected subsets of ...
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### Locally compact groups

Let $S= G_1\bigcup G_2$, where $G_1$ and $G_2$ are two groups. If $S$ is locally compact, is it true that either $G_1$ or $G_2$ is locally compact? Thank you.
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### Prove: If H and G/H are totally disconnected then G is also totally disconnected

Let $G$ be a topological group and $H$ a subgroup of $G$. If $H$ and $G/H$ are totally disconnected, then $G$ is also totally disconnected. With 'totally disconnected' we mean the every connected ...
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### Choosing a canonical fundamental domain

I have a set of equations that partitions a certain space into equivalent regions. For a given point $p$ contained in region $R_1$, there are equivalence relations giving its equivalent position in ...
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### about a topological group

I can't solve this exercise from topology Munkres page 172: Let $G$ be a topological group. (a) Let $A$ and $B$ be subspaces of $G$. If $A$ is closed and $B$ is compact, show that $A\cdot B$ is ...
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### Existence of particular open subgroups, given a profinite group

I have currently read a proof (existence of sections for pro-finite groups (in the book profinite groups of Ribes)) and I did not understand the following two facts used (without mentioning any ...
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### Polish topological group

A friend asked me to help him prove that the topological group $\mathrm{Homeo}(0,1)$ (homeomorphism of $(0,1)$ with the compact open topology) is Polish (that is, separable and completely metrizable). ...
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### Is there any standard terminology for the quotient of a topological group by the connected component of the identity?

If $G$ is any topological group, then the connected component of its identity is a closed normal subgroup $H$. It follows that $G/H$ is a totally disconnected topological group. Often, $G$ will be ...
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### Group Structure on $\Bbb R$

$(\Bbb R,+)$ is a topological group. Is there any other group structure on $\Bbb R$ such that it is still a topological group and this group is not isomorphic to $(\Bbb R,+)$ ? Refer to ...
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### Non-isomorphic Group Structures on a Topological Group

Which Topological Groups Have a Unique Group Structure (up to isomorphism)? I know that there are many non-isomorphic finite groups of same order, so there are many group structures possible for ...
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### Constructing Topological Groups [closed]

In general, is there a way to construct topological groups? That is, given two topological groups $X$ and $Y,$ can I construct a topological group $Z$ using $X$ and $Y$ in said construction? I have ...
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### Given a basis for $\mathbb{R}$, show that it constructs the standard topology on $\mathbb{R}$

Let $q_1, q_2, ...,$ be the rational numbers enumerated. Consider the countable collection $$\mathcal{B} = \{ B_{\frac{1}{n}}(q_i) \ | \ i,n \in \mathbb{N} \}$$ of open balls centered at rational ...
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### Find a locally compact space $X$ with a subspace $A$ that is NOT locally compact.

I'd like to find a locally compact space $X$ with a subspace $A$ that is NOT locally compact. As from here, I know that if $A$ is closed and $X$ is Hausdorff, then $A$ is locally compact. Anyone ...
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### Closed subspace of a compact topological space is compact

Let $X$ be a compact topological space, and $A$ a closed subspace. Show that $A$ is compact. How does this look? Proof: In order to show that $A$ is compact. We need to show that for any open ...
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### Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact.

Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact. This seems pretty straight forward but how will I be able to prove this? I saw this ...
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### What topological group is $\mathbb R/\mathbb Z$?

The integers $\mathbb Z$ are a normal subgroup of $(\mathbb R, +)$. The quotient $\mathbb R/\mathbb Z$ is a familiar topological group; what is it? I've found elsewhere on the internet that it is ...
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### Neighbourhood base about a point $p$ of a topological group

I am reading topological groups from Van der Waerden. The conventions followed in this book are these. An open set that contains the point $p$ is called an open neighbourhood of $p$. Any set which ...
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### Homeomorphism between Space and Product

Do there exist examples of non-empty, infinite spaces X not equipped with the discrete topology for with $X \cong X \times X$?
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### Set of $x$ such that $h \mapsto hx$ is proper

Let $X$ be a locally compact second countable space, and $G$ a locally compact second countable group wich operates continuously on $X$. If $x \in X$, let $\rho_x : g \mapsto gx$. I would like to know ...
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### Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
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### Quotient group $G/G_0$ in Group Topology

I'm stuck on this (apparently) simple thing: If $G$ is a topological group and $G_0$ is the connected component of $G$ containing the identity then $G/G_0$ is discrete if and only if $G_0$ is open. ...
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### Homomorphism Theorem

Let $f$ be an open homomorphism from a topological group $G$ onto a topological group $H.$ We denote $K=Ker(f).$ How can I prove that $\bar f:G/K→H$ is a homeomorphism? I tried to prove it is ...
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### Why is the weak* topology not in general metrizable?

A Banach space is a topological group under addition. The dual is a topological group under the weak$^*$ topology. The weak$^*$ topology is weaker than the operator norm topology, so is it ...
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### A question on Cauchy sequence in topological abelian group

Let $G$ be a topological abelian group. Recall that a Cauchy sequence $(x_n)$ in $G$ is defined to be a sequence such that for any neighborhood $U$ of $0$, there exists an integer $N$ with ...
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### E Hausdorff topological space, G acts properly discontinous

Let $E$ be a Hausdorff topological space, $G$ a homeomorphism group that acts on $E$ properly discontinous, i.e. $\forall e\in E$ exists a neighborhood $U$ of $e$ such that $gU\cap U = \emptyset$ for ...
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### A question on a countable discrete closed set

Let $X$ be a topological group and let $D$ is a countable discrete closed subset of $X$. We also let $\mathcal U= \{U_d: d\in D\}$ of open sets of $X$ such that witnesses that $D$ is closed discrete, ...
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### A question on the right translation

Here is a claim: Let $G$ be a right topological group and $g$ be any element of $G$. Then the right translation $R_g$ of $G$ by $g$ is a homeomorphism of the space $G$ onto itself. How can I ...
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### Combining the axioms of a topological group

According to Wikipedia, a topological group $G$ is a topological space and a group, such that the functions $$(x,y) \mapsto x\cdot y\\x\mapsto x^{-1}$$are continuous. Is the single requirement that ...
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### Action of discrete subgroups E(n) on $\Bbb{R}^n$

Isometry group of euclidean space $\Bbb{R}^n$ is displayed by E(n). We say that a subgroup G of E(n) is discrete if and only if the subspace topology (from E(n)) on G is discrete. If X and Y are ...
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### What is an awesome book as an introduction to hyper groups

I'm a grad studen and i'm choosing an area to follow on my doctorate (in?) and I've been thinking about extension of topological group theory results to topological hypergroups, but for that i need to ...