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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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Show that every topological group is $T_3$

I know that it is sufficient to show that for one point ($e$) and any neighbourhood $U$ of $e$, we have a neighbouhood $V$ with $\bar{V} \subseteq U$. Since $x \to x^{-1}$ is continuous, it follows from continuity of multiplication that we have…
user141592
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Topology on the group of autohomeomorphisms

I am wondering whether the group of all autohomeomorphisms of a compact metric space can be given a reasonable topological group structure? (Preferably, can it be turned into a locally compact group?) I think that moral reason should be no, but I…
TMK
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Generating sets for topological groups

Let G be a compact topological group. Suppose G has a subset X and a normal subgroup N such that the subgroup generated by X is dense in N. Moreover, suppose G has a subset Y such that the subgroup generated by the canonical image of Y in G/N is…
da Fonseca
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The minimal divisible extension

For a prime number $p$, $F_{p}$ is the p-adic number groups and $J_{p}$ is the p-adic integer groups. Is $F_{p}$, the minimal divisible extension of $J_{p}$?
Aliakbar
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Compact open subgroup of a locally compact abelian group

Let $L$ be a compact open subgroup of locally compact abelian group $G$. Is $nL=\{nx;x\in L\}$ an open subgroup of $G$?
Aliakbar
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Group topology and free topological groups

The following result is in Topological Groups and Related Structures, Arhangel'skii, Tkachenko; Notation: $F_a(X)$ denotes the free group on $X$ and $F(X)$ the free topological group on $X$. Claim: Let $f:X\to Y$ be a contiuous mapping of Tychonoff…
user74411
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Example of a finite, non-abelian group in which left invariant metric is also right invariant

I need an example of a finite, non-abelian group $(G, \cdot)$ which satisfies the following condition: If $d$ is a metric on $G$ such that $d(ax, ay)=d(x,y), \ \ \ \ \forall a,x,y \in G$, then $d(xa, ya)=d(x,y), \ \ \ \ \forall a,x,y \in G$. Could…
Sam
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What is a complete topological group?

Given a metric space $ X $, one can forms its completion $ \hat{X} $. However, I have seen that one can also define completeness for topological groups. Can someone explain to me (as simply as possible) what complete topological groups are? I would…
user564167
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Some statements I do not understand when studying topological groups (specifically Korovin functions)

I've been studying the korovin functions for two days. There are 3 statements that I do not know because they are true and I hope you can help me. thanks for your help. First I give you the definition of function of korovin and the context where I…
Luis Prado
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Why are topological groups semitopological groups?

As part of a casual self-study of topology, I have started fooling around with topological groups. I noticed that the article on Wikipedia mentioned that "weakening the continuity conditions" gives the definition for a semitopological group as one…
Will Dana
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Induced action of topological groups

Let $G$ be a polish group, $H$ be an open subgroup of $G$ and $X$ be any metric space on which $G$ act. I want to show the following fact: If the restriction to $H $of the action of $G$ on $X$ is continuous, then the action of $G$ on $X$ is…
Serges
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In topological groups. Is every neighborhood of $e$ supset of a square of a symmetric neighborhood of $e$?

Let $G$ be a topological group, $U$ is a neighborhood of $e$ which is the unit element of $G$. My question is does there exist a neighborhood $H \subseteq U$ of $e$ s.t. $H^{-1}=H$ $H\cdot H\subseteq U$? I have tried to found such a $H$ is a…
Popopo
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Is operator open at topological groups?

Let $(G,\cdot,\mathscr{T})$ be a topological group, then $\cdot$ is indeed continuous, but is it open(close) mapping? It is true at $(\mathbb R,+,\mathscr{T}_{Ord})$, so I guess it is also true in general.
Popopo
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Quotient group is complete, so is the group

Let $G$ be a topological metrizable group and $K$ a normal subgroup of $G.$ Consider the homogeneous space $G/K$ and assume that both $K$ and $G/K$ are complete. I need to prove $G$ is complete. More specifically, assume there is a right-invariant…
William M.
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topological closure of normal subgroup

Let G be any topological group. Let N be the normal subgroup of G. Is it true that closure of N normal? I know the definition of topological group and have done for subgroup but i dont have idea where i should start for normal case.
user195218
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