Tagged Questions
4
votes
1answer
40 views
Sum of Cauchy sequences in an abelian topological group (first countability hypothesis)
We know that, given a first countable abelian topological group $G$, the sum of two Cauchy sequences gives yet another Cauchy sequence (see, e.g., this answer).
For those wondering, we say that a ...
5
votes
1answer
191 views
$G$ 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 ...
2
votes
1answer
43 views
On compact topological group
Must a compact topological group be metrizable? If not, is it separable?
Thanks for any help.
1
vote
1answer
47 views
An example for a non-precompact minimal topological group.
Do you have an example of a non-precompact minimal topological group?
A topological group $(G,\mathcal T)$ is said to be minimal iff it is Hausdorff and for any compatible Hausdorff topology ...
0
votes
3answers
44 views
group of homeomorphisms subgroup
(a) Let X be a topological space. Prove that the set $Homeo(X)$ of homeomorphisms $f:X \to X$ becomes a group when endowed with the binary operation $f \circ g$ .
(b) Let $G$ be a subgroup of ...
4
votes
2answers
54 views
Maximal compact subgroups of $GL_n(\mathbb{R})$.
The subgroup $O_n=\{M\in GL_n(\mathbb{R}) | ^tM M = I_n\}$ is closed in $GL_n(\mathbb{R})$ because it's the inverse image of the closed set $\{I_n\}$ by the continuous map $X\mapsto ^tX X$. $O_n$ is ...
12
votes
1answer
211 views
Every Tychonoff space is an image of a moscow space under a continuous open mapping.
Every Tychonoff space is an image of a moscow space under a continuous open mapping.
A space $X$ is called Moscow if the closure of every open set $U\subset X$ is the union of a family of ...
5
votes
1answer
74 views
A question about quotient under group action
Let $X$ be a Hausdorff space, and $G$ a group acting on $X$ by homeomorphisms. Let $H$ be a normal subgroup of $G$. Is it true that $X/G$ is homeomorphic to $(X/H)/(G/H)$ ?
If so, can you please ...
6
votes
1answer
137 views
Question about pointwise canonically weakly pseudocompact space.
A point $x$ of a space $X$ is said to be a point of canonical weak pseudocompactness if the following condition is satisfied:
For every canonical open subset $U$ of $X$ such that ...
2
votes
1answer
36 views
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 $$ ...
5
votes
2answers
117 views
Are $\Bbb R$ and $\Bbb C$ the only connected, locally compact fields?
I heard that $\Bbb R$ and $\Bbb C$ are the only connected, locally compact fields.
Does anyone know a proof for this result?
3
votes
3answers
129 views
what are all the open subgroups of $(\mathbb{R},+)$
I am not able to find out what are all the open subgroups of $(\mathbb{R},+)$, open as a set in usual topology and also subgroup.
5
votes
1answer
61 views
If $H$ and $G/H$ are compact, then $G$ is compact.
Suppose that $G$ is a topological group and that $H$ is a subgroup of $G$ so that $H$ and $G/H$ are compact. I am trying to show that $G$ must be compact.
The first idea is to use the natural map ...
1
vote
1answer
120 views
Topological Group, symmetric neighborhood, Hausdorff, disjoint open sets
Let $G$ be a topological group with identity $e$. If $A, B$ are subsets of $G$, we let $A * B$ denote the collection of elements $a * b$ for $a \in A, b \in B$, and we let $A^{-1}$ denote the set of ...
1
vote
2answers
64 views
How is the general linear group a topological group?
How to see if the general linear group GL($n$), of non-singular $n$-square matrices over the real (or complex) numbers under matrix multiplication, is a topological group? How to show that matrix ...
4
votes
1answer
76 views
Action of a subgroup of finite index on a tree induced by an action of a group on a tree
Let $G$ be a group wich acts on a tree $\Gamma$. Then $U$ acts on $\Gamma$ for every $U\leq G$.
Question: Why does the following hold?
If $|G:U|<\infty$. Then the minimal $U$-invariant subtree ...
1
vote
1answer
28 views
question about $Y$-homogeneous spaces.
A subspace $Y$ of space $X$ is $h$-dense in $X$, if $Y$ is dense in $X$ and, for each $x\in X$, there exists a homeomorphism $h$ of $X$ onto itself such that $h(x)\in Y$. in this case we say that $X$ ...
3
votes
0answers
102 views
Moscow space-Examples
A space $X$ is called $Moscow$, if for each open subset $U$ of $X$, the closure of $U$ in $X$ is the union of a family of $G_{δ}$ -subsets of $X$ .
For example, Every first countable $T_1$ ...
0
votes
0answers
68 views
Covering space (Lie groups and their maximal tori)
Let $ G $ be a compact Lie group and $ T $ a maximal torus in $ G $. We define the Weyl group $ W $ as the quotient space $ {N_{G}}(T)/T $, where $ {N_{G}}(T) $ is the normalizer of $ T $ in $ G $. We ...
6
votes
0answers
71 views
topological group operation vs homotopy group operation
Let $X$ be a topological group. Let $\tau_1$ and $\tau_2$ representing elements of $\pi_n(X)$. Is it true that
$$ [\tau_1] [\tau_2] = [\tau_1 \tau_2] $$
in $\pi_n(X)$?,
where of course "$[\tau_1] ...
3
votes
1answer
143 views
Metrizable group
Let $ G $ be a metrizable group. If (i) $ K $ is a closed normal subgroup of $ G $ and (ii) both $ K $ and $ G/K $ are complete, then $ G $ is complete.
Here is how I am proceeding:
It can be ...
2
votes
1answer
120 views
A couple of questions about closed subgroups of a topological group.
I am reviewing my previous exams, and I completely missed the following two-part question. It deals with closed subgroups of topological groups under certain situations. I am having trouble working it ...
4
votes
1answer
163 views
Quotient of a locally compact Hausdorff space by a proper action is Hausdorff
I am trying to prove the following:
Let $G$ be a topological group acting properly on a Hausdorff locally
compact space $X$, i.e. preimages of compacts sets by the map
$$G\times X\to X\times ...
9
votes
2answers
195 views
How to show that topological groups are automatically hausdorff?
On page 146, James Munkres' textbook Topology(2ed),
Show that $G$(a topological group) is Hausdorff. In fact, show that if $x \neq y$, there is a neighborhood $V$ of $e$ such that $V \cdot x$ and ...
0
votes
1answer
132 views
Stone-Cech compactification
Is the following statement true or not?
A locally compact Hausdorff space $X$ is a group if and only if its Stone-Cech compactification $\beta X$ is a group.
Thanks.
10
votes
1answer
208 views
Is $[0,1]$ a topological group?
Can one endow the unit interval $[0,1]$ with a group operation to make it a topological group under its natural Euclidean topology?
11
votes
1answer
261 views
Given a group $ G $, how many topological/Lie group structures does $ G $ have?
Given any abstract group $ G $, how much is known about which types of topological/Lie group structures it might have?
Any abstract group $ G $ will have the structure of a discrete topological group ...
1
vote
1answer
83 views
Gillman-Jerison Theorem
How can i prove it?
[Gillman and Jerison] If a dense subspace $Y$ of a Tychonoff space $X$ is $C-embedded$ in X, then $Y$ is $ G_{\delta}-dense $ in $X$.
2
votes
1answer
82 views
discrete subgroup of locally compact abelian group
Let $G$ be a locally compact abelian infinite group but non-compact.
In some paper, the author claims that the dual group $\widehat{G}$ contains an infinite discrete group $K$.
What do you think ...
1
vote
2answers
44 views
clopen subgroup of a topological group
could any one just give hint for this one?
$G$ be a topological group such that $\forall x\in G, x\mapsto xy$ Homeomorphism, $H$ is a open subgroup of $G$, we need to prove $H$ is also closed
1
vote
1answer
84 views
topological groups basic facts
How to show that the usual metric with the usual addition is a topological group?
Can anybody please explain me briefly about topological groups and the way that I need to approach to this question?
2
votes
1answer
72 views
If the action of a group $G$ on $\mathbb{R}$ is properly discontinuous then G is isomorph to $\mathbb{Z}$?
Let $G$ be a topological group, acts on a topological space $X$, such that the map $f: G \times X \rightarrow X:(g,x)\mapsto g*x$ is continuous.
We say that this action is $properly\;discontinuous$ ...
0
votes
0answers
67 views
G is a topological group acts on topological space $X$, is $f_{g}:X\rightarrow X, x\rightarrow g*x$ continuous?
Let $G$ be a topological group acts on the topological space $X$, for an elememt $g\in G$, let's define the map $f:X\rightarrow X, f(x)=g*x$.
I am trying to find if $f$ is continuous?
my best ...
0
votes
1answer
44 views
$G$ finite group acts freely on top. sp. $X$, can we find for every $x\in X$ an open neighborhood such that:
Let $G$ be finite topological group, and acts freely over the hausdorff topological space $X$, i want to prove that every element $x$ in $X$ has an open neighborhood $U_x$ such that:
$g\star ...
8
votes
2answers
251 views
Why isn't there interest in nontrivial, nondiscrete topologies on finite groups?
A topology on a group is required to be compatible with the group structure (multiplication must be a continuous map $G\times G\to G$ and inversion must be continuous). I've only ever seen the ...
5
votes
1answer
78 views
Topological structure of a quotient of ${\rm{SU}}(2)\times{\rm{SU}}(2)$
I'm trying to understand the topology of the product of two three dimensional spheres $\mathbb{S}^3\times \mathbb{S}^3$ quotiented by the action of $\pm 1$ sending a pair of points $(x,y)$ to the ...
4
votes
1answer
91 views
Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group?
And what else can be said, if so?
In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. (It also has a two-sided ...
3
votes
2answers
70 views
open subsets in topological groups
I'm starting to study topological groups, and I noticed that Every single theorem in topological groups I have to use the following statement:
Let $G$ be a topological group and U an open subset of ...
1
vote
4answers
139 views
The real line with its additive group is a topological group?
Maybe it's a stupid question, I'm starting to study topological groups, I'm struggling to prove that the real line is a topological group with its additive group structure and Euclidean topology, ...
4
votes
1answer
98 views
Why are locally compact groups Weil complete?
Why are locally compact groups Weil complete?
Note: A topological group $G$ is Weil complete if every left Cauchy net in $G$ is convergent.
Thank you, and sorry if I have bad writing.
3
votes
1answer
60 views
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 ...
7
votes
1answer
152 views
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 ...
3
votes
1answer
109 views
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 ...
2
votes
1answer
277 views
Why is this quotient space not Hausdorff?
I am trying to show that the following space is not Hausdorff. Consider the topological space $S^1$, and let $r$ be an irrational number. Consider the action of $\mathbb{Z}$ on $S^1$ given by
$$
...
1
vote
0answers
112 views
Identity component of a Lie group
Could any one help me to solve this problem?
Let the identity component $G_0$ of a Lie Group $G$ be the connected component of the identity element $e\in G$. Let $\mu$ and $i$ be the multiplication ...
4
votes
1answer
93 views
Continuous Actions and Homomorphisms
I am learning about the compact-open topology and have a small proposition I am struggling to prove. Let $G$ be a topological group, $X$ a compact, Hausdorff space, and $H(X)$, the homeomorphisms of ...
4
votes
1answer
148 views
Homeomorphisms of X form a topological group
So I'm just learning about the compact-open topology and am trying to show that for a compact, Hausdorff space ,$X$, the group of homeomorphisms of $X$, $H(X)$, is a topological group with the compact ...
1
vote
0answers
118 views
topological group
Recently I'm interested in this open question:
Must every star compact topological group be countably compact?
star compactness ( which implies pseudocompactness ) = for every open cover $U$ of ...
2
votes
0answers
139 views
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 ...
0
votes
0answers
203 views
Properties of Topological Groups
I'm working though William Basener's Topology and Its Applications and I have come across a problem I can't solve. The book defines a topological group as a group equipped with a topology where for ...
