# Tagged Questions

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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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Let $G$ be a finite dimensional real Lie group, and take a bounded ball $B_R(e) \subset G$ in it, coming from the Riemannian metric, which itself is induced from an inner product on $\mathfrak{g} ... 5answers 448 views ### Can$S^2$be turned into a topological group? I know that$S^1$and$S^3$can be turned into topological groups by considering complex multiplication and quaternion multiplication respectively, but I don't know how to prove or disprove that$S^2$... 0answers 95 views ### Measurable function implies equivalent to an exponential function. This is a follow up to this question. In that question, I answered that an exponential function can be uniquely determined by three properties: a functional equation, a weak continuity assumption, and ... 1answer 84 views ### Profinite topology of a Group Let$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$is residually ... 2answers 50 views ### Showing that every path can be well-divided? Let$\gamma: [0,1] \rightarrow S^1$be a path. We'll say that$\gamma$is well-divided if there are$a_1,...a_n$such that:$a_1=0$,$a_n=1\forall_{1\leq i < n}: a_i<a_{i+1}$... 2answers 71 views ### Closure of a topological group Let$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$. But I am not sure ... 1answer 76 views ### Topology of$GL_n(K)$I need to show any of the following results: Consider$K=\mathbb{R}$or$\mathbb{C}$, then, 1) The compact-open topology and the usual topology of$GL_n(K)$are the same. 2) Taking inverses and ... 0answers 21 views ### Circular jump progrssion in a circular pond, lotus petals are arranged along the perimeter. a frog leaps from one petal to another in such a way that starting from a petal, it skips one petal and jumps to the next one; then ... 2answers 34 views ### Showing$U$open in topological group$G\impliesgU$is open If$G$is a topological group, and$U$is an open set in$G$, then do we have that$gU$is also open in$G$? I know that since$G$is a topological group, the mappings$\mu: G^2 \rightarrow G$s.t. ... 0answers 77 views ### Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group. Let$ G $be a locally compact abelian (LCA) group and$ \widehat{G} $the Pontryagin dual of$ G $, i.e., the set of all continuous homomorphisms$ G \to \mathbb{R} / \mathbb{Z} $. Clearly,$ ...
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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 ...
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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?
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### What's so cool about local compactness?

As I study more algebraic number theory, I hear more and more often about local compactness: locally compact fields, locally compact topological groups, Stone-Čech compactification of locally compact ...
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### Discrete subgroups of isometry group $\mathbb{R}^n$

Let $G$ be a Hausdorff topological group. We say that a subgroup $S$ of $G$ is discrete if and only if the subspace topology (from $G$) on $S$ is discrete. Note that isometry group of euclidean space ...
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### Neighborhood in topological groups

Let $G$ be a topological group, $e$ the neutral element and $U$ a neighborhood of $e$. Claim: Then there exists a neighborhood $V$ of $e$, such that $V^2 \subseteq U$. This should follow easily from ...
I have some general questions around the profinite topology on a group $G$. On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that The profinite topology on a group is ...
The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...