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Questions tagged [topological-groups]

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.

A topological group is a group endowed with a topology such that both the group operation and inversion are continuous maps. Every group can be understood as a topological group, if we take the discrete topology.

Topological groups are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis, see e.g. Pontryagin duality.

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$G$ topological group. $G=UD=DU$, $D$ dense, $U$ open set

Let $G$ a topological group, $U \neq \emptyset$ open subset of $G$, $D$ dense in $G$. Prove $G=DU=UD$. Attempt 1 Take $V=U \cap U^{-1}$. $V$ is open set. Let $x \in G$ then $xV$ is open and $D$ dense, so $xV \cap D \neq \emptyset$. Then exists $v…
PSW
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For any open set $U$ of $G$ we have $(U \cap G_0)G \subseteq \bigcup \{X \cap Y | X,Y \in \Omega(G), XY^{-1} \subseteq U \}$

I'm starting to learn about etale groupoids and I have this specific question about topological groupoids. It should be quite simple but I'm having trouble showing it. Note $G_0$ is denoting the units in $G$. I start with a point $x \in (U \cap…
Chi
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Proving that $G/H$ is Hausdorff

Let $G$ be a topological group and $H$ be a normal subgroup. I want to show that $G/H$ is a topological group. I have managed to prove that $m:G/H\times G/H\to G/H, \bar a \bar b\to \bar{ab}$ and $i:G/H\to G/H: \bar a\to \bar {a^{-1}}$ is…
Runyang Wang
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Are the left uniformly continuous functions on a topological group dense in $L^1(Haar)$?

Let $G$ be a locally compact second countable topological group (Lie group if it helps). A function $f : G \to \mathbb{C}$ is left uniformly continuous if for all $\epsilon > 0$ there exists an open neighborhood $U$ of the identity such that…
someone
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$GL_n(\mathbb{R})$ is open in $M_n(\mathbb{R})$ but not closed?

It is well known that for any subgroup $F$ of a topological group $G$. If $F$ is open in $G$, then it must be closed $G$. However, we know: $GL_n(\mathbb{R})$ is open in $M_n(\mathbb{R})$ $GL_n(\mathbb{R})$ is a subgroup But we also know…
Hamilton
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Every discrete subgroup of a topological $T_1$ group is closed.

Let $G$ be a topological $T_1$ group. If $H$ is a discrete subgroup of $G$, then there exists $U\in V(1)$ (here $V(1)$ is a filter of all neighborhoods of $1$) such that $U\cap H=\{1\}$. Let $V\in V(1)$ such that $V^{-1}V \subset U$. Note…
user990165
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Proving that if $xy\in W$ then $\exists U,V\subseteq G:U\cdot V \subseteq W$

I'm having some trouble proving the following proposition:~ Let $(G,\cdot)$ be a topological group, $x,y\in G$ and $W$ an open neighbourhood of $x\cdot y$. Then, there are open neighborhoods $U$ and $V$ of $x$ and $y$ respectively such that$$U\cdot…
Eduardo Magalhães
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A question on topological group

Let $G$ be a topological group. We know that if $A$ is a closed subset of $G$ and $B$ a compact subset of $G$, then $A+B$ is a closed subset of $G$. My question: Is the above statement true whenever $B$ is a $\sigma-$compact set of $G$? Thank you
Aliakbar
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What needed to be checked for a topological subgroup?

I have the following Problem: let $G$ be a topological group. We denote by $G_0$ the path connected component of the identity and by $G_0'$ the connected component of the identity. I now want to show that $G_0$ is a normal subgroup of $G$ I wanted…
user123234
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A set $U$ is neighborhood of $g$ iff $g^{-1}U$ is neighborhood of identity element.

I have some difficulty proving the following fundamental statement on the topic of topological groups; a topic that I have just started to study independently. Let $G$ be a topological group. Consider an arbitrary element $g$ of $G$. It follows that…
Julius Monk
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Does open subgroup of topological group contains identity component

I am asked to prove that if we have a topological group $G$ then if $H$ is open subgroup of $G$ we have that $H$ contains identity component of $G$. I dont see how those two relates. I know that identity component of $H$ is containted in identity…
MotionMath
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Proof of $\overline{A} = \bigcap_{U\in\mathfrak{U}}(A + U) = \bigcap_{U\in\mathfrak{U}}\overline{A + U}$

Proposition. Let $G$ be an abelian (probably not Hausdorff) topological group, let $A$ be a subset of $G$, and let $\mathfrak{U} = \{U\subseteq G \mbox{ : neighbourhood of }0\}$. Then, the closure $\overline{A}$ is written as $$ \overline{A} =…
K. Y.
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Compactly supported function on a topological group is uniformly continuous

Let $G$ be a topological group, and $f: G \rightarrow \mathbb C$ function. We say $f$ is uniformly continuous if for any $\epsilon > 0$, there exists a neighborhood $W$ of $1_G$ such that $xy^{-1} \in W$ implies $|f(x) - f(y)| < \epsilon$. We say…
D_S
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Openness preserved for product with normal subgroup

For a group $P$ with open subgroup $Q$ and a group $N$ such that $P$ normalizes $N$ and $P\cap N$ is closed in $P$, I wonder whether $QN$ is an open subgroup of $PN$.
Wembley Inter
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Can we Classify the Homomorphic Images of Tori?

Is it known for which topological groups $X$ there exists a positive integer $d$ and a surjective, continuous group homomorphism $([0,1]\text{ mod } 1)^d \to X$? Certainly, any such $X$ must be compact and path connected. Also, it's set of…
Thomas Winckelman
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