A topological group is a group endowed with a topology such that the group operation and inversion are continuous maps. They are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis.

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support compact modulo subgroup

I am studying (co)-induced representations of topological groups and I came across the following situation: $G$ is a topological group, $H$ a closed subgroup and $f\colon G\to W$ a set-theoretic map, ...
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29 views

LCA - groups under continuous homomorphisms

can someone help me out with this question? LCA stands for Locally compact Hausdorff abelian group. The question is posted in the attached image Let $A$ and $B$ be LCA-groups and $H$ a (not ...
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Equivalence to being a topological group

Just some notation I am using: A topological group $G$ is a group with a topology such that $o : G^2 \to G : (x,y) \mapsto xy$ and $inv : G \to G : x \mapsto x^{-1}$ are continuous in the ...
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31 views

Why the multiplicative group $G_m$ is called a 1 dimensional torus?

I am reading a definition saying that an algebraic group over a field $K$ is called a torus if it is isomorphic to product of copies of the multiplicative group $G_m = K^*$. I don't understand why ...
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1answer
25 views

What prevents the restriction of a Haar measure to a closed subgroup from being a Haar measure?

Let $\mu$ be a Haar measure on a locally compact Hausdorff topological group $G$, and let $H$ be a closed subgroup of $G$. If we restrict $\mu$ to the Borel sets of $G$ which are contained in $H$ ...
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1answer
23 views

Showing that compact metrizable abelian group has enough characters

I wanted to know if anybody has some clue as why a compact metrizable abelian group has enough characters to sepatare points. (a character on a topological group is a continuous homomorphism from the ...
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25 views

universal bundle of topological monoids

Let $M$ be a topological monoid. There is a classifying space $BM$ (cf. canonical map of a monoid to its classifying space). When $M$ is a group $G$, there is a principal $G$-bundle $EG\to BG$ such ...
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When does a topological space inherit multiplication from a dense subspace?

Suppose $K$ is a compact topological Hausdorff space with a dense subspace $G$. Moreover, let $G$ have a group structure which is compatible with the topology inherited from $K$. i.e. $G$ is a ...
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1answer
46 views

Is a compact subset of a topological group G/N closed if G is hausdorff?

I just used this hypothesis when I was proving a theorem.But I was not sure if this hypothesis is correct.
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1answer
37 views

Compact paratopological groups are automatically topological groups.

A compact paratopological group is a topological group. How to prove it? An abelian paratopological group is a topological group. Is this right? A paratopological group is a topological semigroup ...
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1answer
23 views

example of a particular topological group

Can someone give an example of a topological group $G$ that is not Hausdorff but that contains a fundamental system of neighbourhoods of $1\in G$ consisting of quasi-compact subgroups? Thanks in ...
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1answer
16 views

Why is there a set $W$ (to be described below) such that $\mathbb{A}_K = W + K$?

To prove the compactness of $\mathbb{A}_{\mathbb{Q}}/ \mathbb{Q}$ (and hence $\mathbb{A}_K/K$ for an arbitrary number field $K$), one finds a set $W \subseteq \mathbb{A}_{\mathbb{Q}}$ of the form $$ ...
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suspension foliations on thickened surfaces

I've seen this statement without proof in a peer reviewed journal and I'm looking for a proof: "If $L$ is an oriented surface with boundary($\neq D^2$), and $C$ is a designated boundary component, ...
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15 views

Is the unitary group of a pre Hilbert space contractible?

for a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for the strong operator topology (Dixmier and Douady, ...
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1answer
25 views

Blichfeldt-Minkowski Lemma

I'm trying to understand a proof of the following result Theorem: Let $K$ be a number field, and $|| \cdot ||$ the idelic norm (product of the normalized absolute values at each place). There ...
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0answers
36 views

Quotient group of $\mathbb{R}^2$ by irrational line

In a section about topological groups, exercise 4.10 in I.M. James' General Topology and Homotopy Theory asks Show that for irrational values of $\alpha$ the factor group of the real plane ...
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1answer
39 views

The whole group is covered by compact translating of subgroups

$G$ is a locally compact (may not necessarily Hausdorff) group, $H$ is a subgroup in $G$, $G/H$ is compact as a quotient space , then there exist a compact subset $K$ such that $G=KH$(or $G=HK$).
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Is there a non-abelian Lie group which is homeomorphic to an $n$-dimensional torus $\mathbb{T}^n$?

I've learned that a compact connected abelian Lie group must be a torus. Of course, conversely, a torus as a group is abelian. I wonder if 'homeomorphic to a torus' is enough to imply abelian. ...
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3answers
27 views

Example for algebraic homomorphism between topological groups which is not continuous

I am quite sure there should be an easy example of: (algebraic) homomorphism between topological groups which is not continuous. However, I do not see one immediately.
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3answers
46 views

Continuous homorphisms between topological groups.

Let $G,H$ be topological groups and let $\rho: G \rightarrow H$ be a homomorphism of topological groups. I understand this to mean that $\rho$ preserves group structure and is a map between the ...
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1answer
24 views

A topological group is embeddble in a product of a family of second-countable topological groups if and only if it is $\omega$-narrow

How to prove the following property: a topological group is topologically isomorphic to a subgroup of the product of some family of second-countable topological groups if and only if it is ω-narrow
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1answer
46 views

Comparing the Samuel and Stone-Čech compactifications of a Hausdorff topological group

Let $G$ be an Hausdorff topological group and let $\beta G$ be the Stone-Čech compactification of $G$. Now, $G$ is also a uniform space with respect to the so-called right uniformity. Let $S(G)$ be ...
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19 views

$ \pi : O(n) \rightarrow O(n)/O(n-k) \cong V_{n,k}(\mathbb{R}) $is a principal $ O(n-k) $-bundle.

I'm trying to prove that $ \pi : O(n) \rightarrow O(n)/O(n-k) \cong V_{n,k}(\mathbb{R}) $; $ A \longmapsto (Ae_1, ... ,Ae_k) $ (the projection from the orthogonal group to the Stiefel manifold) is a ...
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31 views

open subgroup of normal topologcial group is normal

$G$ a topological group and $H⊆G$ an open subgroup, $G$ is normal iff $H$ is normal. Remark: Here, G is not necessarily a Hausdorff space. A topological space $X$ is a normal space if, given ...
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1answer
65 views

Is the completion of a metrizable topological group metrizable?

Let $G$ be a topological group and its two-side uniformity $\mathcal{U}$ (that is the uniformity generated by right uniformity and left uniformity of $G$) coincides with the uniformity of a metric ...
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1answer
99 views

Does every continuous action of $S^1$ on $R^n$ have a fixed point?

I certainly can't think of one that doesn't. I am aware that there are decompositions of $R^n$ as a union of embedded $S^1$'s, but none of these seem like they would support a continuous action. ...
3
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1answer
73 views

Is there such a norm on any totally disconnected local field?

Let's set this definition of local field: Let $\mathbb{K}$ be a field and a topological space (non-discrete and totally disconnected). Then $\mathbb{K}$ is called a local field if both ...
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0answers
9 views

Show that representative functions on a profinite group factors. [duplicate]

Let $G$ be a compact group. A representative function $f\in\mathcal{C}(G,\mathbb{K})$ is a function such that $\dim\left(\operatorname{span}\left(Gf\right)\right)< \infty$. Remark that the ...
3
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1answer
23 views

Compactness of a group with a bounded left-invariant metric

Let $G$ be a group equipped with a left-invariant metric $d$: that is, $(G,d)$ is a metric space and $d(xy,xz) = d(y,z)$ for all $x,y,z \in G$. Suppose further that $(G,d)$ is connected, locally ...
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40 views

how do i prove the existence of this norm?

I am reading an article that states: Let $\mathbb{K}$ be a fixed local field. Then there is an integer $q=p^r$, where p is a fixed prime element of $\mathbb{K}$ and r is a positive integer, and a ...
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1answer
32 views

Proof of a result on closed subgroups of Galois group

Let $M\supseteq K$ be an algebraic normal extension. Then its Galois group is profinite. I have been told that in this hypothesis, if $H\leq G$, then $H''=H\iff \bar H=H$, where ...
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Is $\mathcal O^{\ast}/U^{(n)}\cong (\mathcal O/\mathfrak p)^{\ast}$ as topological groups?

I have the following: $K$ is a field with discrete valuation $v$, $\mathcal O$ its valuation ring and $\mathfrak p$ the maximal ideal and $U^{(n)}=1+\mathfrak p^n$ the $n$-th unit group for $n\geq 1$. ...
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2answers
135 views

$\epsilon$-dense subsets on $\mathbb R/\mathbb Z$.

Let $\langle M, d\rangle$ be a metric space. We say that $A \subset M$ is $\epsilon$-dense if every open ball of radius $\epsilon$ contains a point of $A$. Now let $T=\mathbb R/\mathbb Z$, the ...
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1answer
32 views

Complex numbers modulo integers

Is there a "nice" way to think about the quotient group $\mathbb{C} / \mathbb{Z}$? Bonus points for $\mathbb{C}/2\mathbb{Z}$ (or even $\mathbb{C}/n\mathbb{Z}$ for $n$ an integer) and how it relates ...
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2answers
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Three questions from σ-compact spaces and topological groups [closed]

every locally compact subgroup of a Hausdorff group is closed. A Hausdorff and $σ-$compact space X is a Baire space if and only if the set of points at which is $X$ is locally compact is dense in ...
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2answers
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Prob. 5 (a) in Supplementary Exercises in Munkres' TOPOLOGY, 2nd ed: How to show that this map is a homeomorphism?

Let $G$ be a topplogical group, and let $H$ be a subgroup of $G$. Let $G / H $ denote the collection of all left cosets of $H$ in $G$, and let $a$ be a fixed element of $G$. Let the map $f \colon G / ...
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1answer
67 views

When a sigma-finite space is a sigma-compact space?

$X$ is a topological space, $m$ is a $\sigma-$finite measure on $B(X)$, and what condition can make $X$ be a $\sigma-$compact space? This question is from topological groups (for me). Locally compact ...
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1answer
16 views

Construction of a given neighbourhood in a locally compact group

Let $G$ be a locally compact group. Why is it possible to select a compact neighbourhood $U$ of $e \in G$ such that $U=U^{-1}$ and $gU^2 \subset V$? This is a construction quickly stated by Helgason, ...
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33 views

When can a group be made into a ring? How `little' of the ring structure must be specified?

Given a (topological) abelian group $G$ and a (bicontinuous) $G$-bilinear map $\mu: G \times G \to G$, clearly $G$ becomes a (topological) ring by specifying $$ x y := \mu(x, y) \quad \forall x, y \in ...
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1answer
25 views

The appropriate translation of Sets of Positive Measure is positive

$A$ and $B$ are two measurable subset of $ \mathbb{R}$, and $m$ is a Lebesgue measure on $\mathbb{R}$, if $m(A)>0$ and $m(B)>0$, then there exist a $ x\in \mathbb{R}$, such that $m(A \bigcap ...
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2answers
29 views

an existence question from topological groups

$G$ is a topological group, $A$ and $B$ are the subsets of $G$, we denote $AB$=$\{ab:a \in A, b\in B \}$. Let $G$ be a locally compact Hausdorff topological group, $m$ is a left Haar measure on $G$, ...
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2answers
34 views

Topology of $\Bbb{Q}_p$

Let $a\in \Bbb{Q}_p$. Is $ a+p^x\Bbb{Z}_p$ an open set around $a$ in the topology of $\Bbb{Q}_p$. Here $x \in \Bbb{Z}$. Also I have another question. Is $\mathbb{Z}_p$ open in $\Bbb{Q}_p$?
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1answer
42 views

intersection about the second category

$G$ is a locally compact Hausdorff topological group, $A$ and $B$ are two Borel subsets of $G$, and $A$ and $B$ are both of the second category in $G$, then there exist an element $x\in G$, such that ...
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Topological group, which is second category in itself, is a Baire space.

A Baire space is a topological space in which the union of every countable collection of closed sets with empty interior has empty interior. $G$ is a topological group, if $G$ is of the second ...
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1answer
69 views

Compact subgroups are contained in open compact subgroups in locally profinite groups

Let $G$ be a totally disconnected, Hausdorff, locally compact group. In the wikipedia page about these groups there is a claim that any compact subgroup of $G$ is contained in some compact open ...
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1answer
25 views

Two questions about increasing unions of compact subsets of a locally compact Hausdorff group.

I have two questions to ask related to my research. Question 1. Let $ G $ be a locally compact Hausdorff group. Is it possible that $ G $ is the union of a chain of compact subsets (ordered by ...
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0answers
34 views

Choice of a dense subset of a separable Banach space

I recently came across the following statement, and still can't prove it: Statement: Suppose $X$ is a separable,closed subspace of $L^1(G)$, where $G$ is a locally compact group. Since $X$ is ...
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31 views

Closed mapping theorem for almost connected groups

Edit: I originally asked if $\phi$ is a closed map. Clearly it isn't in general, as the real numbers can be mapped to the torus with dense image. Let $G$ and $H$ be connected locally compact groups, ...
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1answer
16 views

Sigma-compact Polish groups

I would like to see an example of a sigma-compact Polish group which is not locally compact. I know that e.g. $l^{\infty}$ is a topological group which is sigma-compact but not locally compact. But ...
2
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2answers
39 views

Any finite index subgroup of $\mathbb Z_p$ is open [duplicate]

I'm trying to show that every finite index subgroup $H$ of $\mathbb Z_p$ is open. Since $H$ has finite index, it is equivalent (and perhaps easier) to show that it is closed. But I've tried showing ...