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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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Component of identity

Could you please help me to solve this one: The connected component of the identity of a topological group is a normal subgroup? I also need a hint to show path-connected components are normal subgroups. I am not familiar with deep properties of…
Myshkin
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Is intersection of connected subgroups connected?

Let $G$ be a compact group. If $A_{\alpha}$, $\alpha \in I$ is a family of closed connected subgroups in $G$, then is it true that $\bigcap_{\alpha \in I}A_{\alpha}$ is connected?
Kiran
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Separability of a group and its dual

Here is the following exercise: Let $G$ be an abelian topological group. $G$ has a countable topological basis iff its dual $\hat G$ has one. I am running into difficulties with the compact-open topology while trying this. Any help?
Chera
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A question related to "Topology induced by the completion of a topological group"

I am sorry if I mistake the answer posted on the question Topology induced by the completion of a topological group. Stated as in that thread, let $G$ be a topological abelian group with a countable fundamental system of neighborhoods,…
user119882
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First isomorphism theorem for topological groups

Let $f$ a continuous homomorphism from a topological group $G$ onto a topological group $H$. We denote $K = Ker(f)$. I already proved that $\overline{f}:G/K\to H$ defined by $\overline{f} (xK)=f(x)$ is an algebraic isomorphism and continuous. Now, I…
user73564
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Compact connected group and torsion-free dual

Here's yet another exercise that stumped me: Let $G$ be a compact abelian topological group. Then $G$ is connected iff its dual $\hat G$ is torsion-free. Any hints/solutions will be appreciated.
Chera
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Definition of topological group

Let $U$ be open in a topological group, G. Why then is it necessarily true that $UH$ where $H$ is some subgroup of $G$ open in $G$? (I think I don't quite get the idea of a topological group even after reading its definition on Wiki. Grateful if…
brian
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Subgroup of the unit circle under complex multiplication

Let $T=\{z:|z|=1 \mathbin{\text{and}} z \in \mathbb C\}$ be the unit circle in the complex plane, considered as a topological group under complex multiplication and the usual topology. Show that every subgroup of $T$ is either dense in $T$ or…
Milan Amrut Joshi
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Connected components of this topological group

I was confused: today some undergraduate asked me the following exercise let $m,n\in \mathbb{Z}_{\geq 1}$ and $$G:=\{(a,b)\in \mathbb{C}^\times \times\mathbb{C}^\times: a^m b^n=1\}$$ as a topological group, with the "usual" topology (as a subgroup…
youknowwho
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Continuous action

Let $G$ be a polish group, $H$ an open subgroup of $G$. Now assume that $H$ acts by isometries (For all $h\in H$, the map $X\ni x\longmapsto X$ is an isometry) and continously on a metric space $(X,\delta)$. We define $$F=\{f:G\longrightarrow…
Serges
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Is the orbit space of a Hausdorff space by a compact Hausdorff group Hausdorff?

Let $G$ be a compact Hausdorff group. Let $X$ be a Hausdorff space. Suppose $G$ acts continuously on $X$. Is the orbit space $X/G$ Hausdorff? If not, I would like to know an counter-example. Remark As my answer to this question shows, if $X$ is a…
Makoto Kato
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Interesting topological groups that aren't manifolds?

Does anyone know of examples of "pathological" topological groups that are actually used? I am particularly interested in infinite non-Hausdorff examples.
Perry Bleiberg
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Are inversion and multiplicaton open?

If $G$ is a topological group, are inversion $G \to G$ and multiplication $G\times G \to G$ open mappings? More concretely, I try to show that division of complex numbers $$\{(z,w) \in \mathbb{C}^2;\; w \neq 0\} \to \mathbb{C},\; (z,w) \mapsto…
k.stm
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Definition of precompactness in a topological group $G$

I have seen that the definition of precompact sets in a topological group $G$ is a bit tricky. Can someone please explain? I saw that it has to do something with totally bounded sets. Is there a more 'natural' definition of precompactness? For…
User666x
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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 that is algebraically a group. In other words, it is…
David Chan
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