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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example of a discontinuous group operation in $\mathbb{R}$, under usual metric

someone has an example of that? I know that if G is a topological connectedness group ( operation continuous under the topology) then every neighborhood containing the identity elemental, generates ...
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The inverses of open sets

I am not sure if this is already posted, though, I hope I can get some help, and thank in advance. This question arises from the proof of the following. Proposition: Let G be a topological ...
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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 ...
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totally disconnected orbit-stabilizer theorem

So I'm aware that the orbit-stabilizer theorem does not hold for arbitrary spaces with a transitive action by a topological group, but I wonder if it works in the following situation. Let $G$ be a ...
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does isomorphic groups induce homeomorphic quotients

suppose $X$ is a topological space and $G$ and $H$ are groups acting on it. 1) if $G$ is isomorphic to $H$ do we have necessarely $X/G$ is homeomorphic to $X/H$ 2) suppose $G$ and $H$ are two ...
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how to factor a map by a group action

Let $X$ and $Y$ be topological spaces and a surjective map $f:X\rightarrow Y $. Suppose that a group $G$ acts on $X$. and let $\pi:X\rightarrow X/G$ be the quotient map. 1) Under what conditions $f$ ...
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Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

Could you please explain me the reason why they are isomorphic? Thanks, bye!
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Is there any relation between a group being unimodular and having equivalent uniform structures?

Recall: A topological group is said to have equivalent uniform structures if its left and right uniform structures coincide. A locally compact group is said to be unimodular if left Haar measures and ...
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Shrinking Group Actions

Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ ...
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If $G$ is a locally compact Hausdorff group, when does $G/Z$ have a probability Haar measure?

I am reading an introductory material about topological groups and the question in the tittle comes up. Due this Proposition Proposition. A locally compact Hausdorff topological group $G$ is ...
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Compact group actions and automatic properness

I am currently re-reading a course on basic algebraic topology, and I am focussing on the parts that I feel I had very little understanding of. There is one exercise in the chapter devoted to groups ...
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Restricted Direct Products in Koch's Number Theory

On p.353 of Number Theory: Algebraic Numbers and Functions by Helmut Koch, he considers a group $G$ which is the restricted direct product of the locally compact abelian groups $G_i$ with respect to ...
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Topology on the general linear group of a topological vector space

Let $K$ be a topological field. Let $V$ be a topological vector space over $K$ (if it makes things convenient, you may assume it is finite dimensional). Naive Question: Is there a canonical way of ...
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Does a topological group need to have a uniformity making all group operations uniformly continuous?

Let $G$ be a topological group. $G$ comes equipped with a left (resp. right) uniformity $\mathscr{L}$ (resp. $\mathscr{R}$) which can be characterized as the coarsest uniformity which is compatible ...
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Why is every discrete subgroup of a Hausdorff group closed?

I have just begun to learn about topological group recently and is still not familiar with combining topology and group theory together. I have read a useful property of discrete group on the ...
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Useful sufficient conditions for a topological space to be the underlying space of a topological group?

Here is a question that I have had in my head for a little while and was recently reminded of. Let $X$ be a (nonempty!) topological space. What are useful (or even nontrivial) sufficient ...
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Intersection of neighborhoods of 0. Subgroup?

Repeating for my exam in commutative algebra. Let G be a topological abelian group, i.e. such that the mappings $+:G\times G \to G$ and $-:G\to G$ are continuous. Then we have the following Lemma: ...
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Is addition continuous?

I'm going to ask a very silly question, so I'm begging you to be understanding if it is absolutely trivial, or if it's an exercise in some Bourbaki. I'm afraid of asking you, because the question ...
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Topological group: Multiplying two loops is homotopic to linking these paths?

Let G be a topological group and let $s_1$ and $s_2$ be loops in G (both loops are based at the identity e of G). Is it true that the loop $s_1s_2$ (where the multiplication is the one of the group ...
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Colimit of topological groups (again)

In Direct limit, Martin rightly pointed out that my naive construction (now deleted) of the colimit (direct limit) of topological abelian groups was wrong. He shows how to do it properly (at least the ...
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what are the product and coproduct in the category of topological groups

I know the limits in the categories of groups, abelian groups and topological spaces and was wondering about the same thing.
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Is a direct limit of topological groups always a topological group?

If $(G_i,f_{ij})$ is a direct system of topological groups, is it always the case that the topological\group-theoretical direct limit $G:=\varinjlim_iG_i$ is a topological group? (The topology on $G$ ...