0
votes
1answer
36 views

Existence of particular open subgroups, given a prof-finite 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 ...
1
vote
1answer
34 views

Existence of a neighbourhood of a compact set ( from james fibrewise topology)

I'm reading James' Fibrewise topology book and I'm trying to understand the proof of proposition 7.4 , it says: Let X be a proper G-space . Then X is fibrewise regular over X/G. Proof For any $x \in ...
5
votes
4answers
525 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
1
vote
0answers
38 views

How can we characterize all topological groups given $G$?

The idea is that all topologies on G (not necessarily making it a topo group) can be completely specified by a set of functions $F = \{f: G \to G\}$ if you form a basis for the topology like: $B = ...
4
votes
1answer
102 views

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). ...
0
votes
1answer
27 views

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 ...
6
votes
0answers
106 views

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 ...
10
votes
2answers
137 views

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 ...
1
vote
1answer
54 views

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 ...
1
vote
1answer
64 views

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 ...
1
vote
3answers
46 views

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 ...
0
votes
1answer
32 views

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 ...
0
votes
1answer
58 views

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 ...
3
votes
2answers
156 views

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 ...
0
votes
1answer
46 views

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 ...
7
votes
2answers
112 views

Homeomorphism between Space and Product

Do there exist examples of non-empty, infinite spaces X not equipped with the discrete topology for with $X \cong X \times X$?
1
vote
0answers
16 views

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 ...
1
vote
0answers
29 views

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 ...
1
vote
1answer
47 views

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. ...
1
vote
1answer
48 views

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 ...
1
vote
1answer
103 views

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 ...
0
votes
1answer
34 views

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 ...
2
votes
2answers
100 views

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 ...
3
votes
1answer
63 views

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, ...
1
vote
1answer
27 views

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 ...
2
votes
1answer
51 views

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 ...
0
votes
1answer
24 views

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 ...
2
votes
3answers
146 views

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 ...
2
votes
1answer
24 views

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?
13
votes
1answer
169 views

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 ...
0
votes
1answer
104 views

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 ...
2
votes
1answer
18 views

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 ...
0
votes
0answers
40 views

Different profinite topologies on a group?

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 ...
0
votes
0answers
47 views

Subgroup Separability translated in Profinite Topology

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 ...
1
vote
2answers
81 views

Why is $\mathbb R^n$ under the Zariski topology not a topological group?

Reasons that $(\mathbb R^n, +, \mathcal Z)$ is not a topological group: Given any two distinct points $\vec{p},\vec q \in \mathbb R ^n$ let $P$ be the unique hyperplane through $\vec p$ which is ...
3
votes
1answer
136 views

Product of totally disconnected space is totally disconnected?

I read that the cartesian product with the product topology of a family of totally disconnected topological spaces is totally disconnected, too. Is that true? How are the connected components in the ...
3
votes
2answers
64 views

Topological Group $G$ totally disconnected $\Rightarrow$ $G$ hausdorff?

On Wikipedia, I read that a topological group is necessarily Hausdorff if it is totally disconnected. Is that true? I read it on this page: http://en.wikipedia.org/wiki/Totally_disconnected_group ...
3
votes
0answers
49 views

Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
2
votes
1answer
44 views

Uncountable dense measurable subgroups of $\mathbb{C}$

Is it possible to have an uncountable proper dense subgroup of $\mathbb{C}$ which is also Baire or Lebesgue measurable?
2
votes
0answers
81 views

Orientability as a topological property

Can one prove that orientability(of a manifold)is a topological property without using algebraic topology? That is, using a combination of general topology,linear algebra,and topological groups(such ...
11
votes
1answer
122 views

Proving that a metric space is a group

I'm stuck on this relatively hard problem. Let $G$ be a non-empty set, $d$ a distance on $G$ and $\cdot$ an associative operation on $G$ $\cdot$ is such that $$\forall a \in G , \forall x \in G ...
0
votes
1answer
115 views

$G$ topological group, $H$ discrete normal subgroup, $p$ projection, form Covering Space.

Let $G$ be a topological group. Let $H$ be a discrete normal subgroup of $G$. Let $p : G \to G/H$ be the projection map. Show that $(G, p, G/H)$ form a covering space. Here is what I have so far: ...
2
votes
1answer
53 views

Finitely generated subgroups of prodiscrete groups

Suppose $(G_n, p_{n+1,n}:G_{n+1}\to G_n)$ is an inverse sequence of discrete groups and (obviously continuous) group homomorphisms. Let $G=\varprojlim G_n$ be the inverse limit (with the usual inverse ...
2
votes
1answer
43 views

Compactness in a short exact sequence of topological groups

Suppose $H,G,K$ are abelian Hausdorff topological groups and $0\to H\overset{\alpha}\to G\overset{\beta}\to K\to 0$ an exact sequence of continuous homomorphisms. If $H$ and $K$ are compact, can we ...
0
votes
1answer
28 views

Does a Normal Moore paratopological group always have countable chain condition?

Does a Normal Moore paratopological group always have countable chain condition? Or is it separable? Thanks for your help.
2
votes
5answers
106 views

Open subgroups of $\mathbb{R}$ [duplicate]

Let $G$ be a nonempty open subset of $\mathbb{R}$ (with usual topology on $\mathbb{R}$) such that $x,y\in G$ implies that $x-y\in G$. Show that $G=\mathbb{R}$. Clearly $0\in G$. Now how to show that ...
0
votes
1answer
41 views

Show that if $X$ is discrete, then $\phi$ is continuous.

Let $S(X,X)$ of all mappings of a set $X$ to itself, taken with the topology of pointwise convergence. Define $\phi: S(X,X) \times S(X,X) \to S(X,X)$ with $\phi((f,g))=f \circ g$. Show that if ...
0
votes
1answer
37 views

Show the multiplication mapping of $S \times S \to S$ is not (jointly) continuous

Let $S=R \cup {\alpha}$ be the one-point compactification of the usual space $R$ of real numbers. Define multiplication on $S$ by the rule $xy=x+y$ if $x$ and $y$ are in $R$, and $xy=\alpha$, ...
1
vote
1answer
91 views

Can a locally compact group with closed singleton be countable but not discrete?

Problem: Prove if a locally compact group $(G,*)$ contains a closed singleton then it must be either discrete or uncountable Proof Given: Assuming $G$ is countable we can write $G = \displaystyle ...
4
votes
0answers
90 views

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 ...