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.

learn more… | top users | synonyms

21
votes
1answer
456 views

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 $\rho : G\times X\rightarrow X$ ...
3
votes
0answers
186 views

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 ...
3
votes
1answer
245 views

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 ...
3
votes
1answer
129 views

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 ...
11
votes
3answers
565 views

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 ...
4
votes
2answers
254 views

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 ...
7
votes
2answers
711 views

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 ...
20
votes
1answer
482 views

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 ...
6
votes
1answer
189 views

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: ...
12
votes
2answers
989 views

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 ...
8
votes
2answers
802 views

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 ...
16
votes
3answers
806 views

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 ...
13
votes
2answers
696 views

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.
12
votes
3answers
764 views

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 $G$ ...