# Tagged Questions

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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### Which Algebraic Properties Distinguish Lie Groups from Abstract Groups?

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group, and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
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### Halmos Measure Thoery section 62 exercise 3

Is there a locally compact group $G$ and a Borel measure $\mu$ on $G$ such that \begin{equation*} H=\{g\in G\mid \mu(gE)=\mu(E) \: \text{for all measurable} \: E\} \end{equation*} is not a closed ...
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### An example of a compact multiplicatively unbounded ring

My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
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### Why is topological group not a popular topic?

In Japan, there are many universities with a formal course about topological group using the classic by Pontryagin. Yet topological group is not studied in a formal course in many other countries, ...
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### The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
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### Prove that the quotient map $P:G \to G/H$ is a covering space.

Let $G$ be a topological group and $H$ is a subgroup of $G$. Suppose that the subspace topology on $H$ is the discrete topology. Prove that the quotient map $P:G \to G/H$ is a covering space. My ...
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### Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell ...
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### The simply-connectedness of quotient space

If $U$ is a Lie group with a closed subgroup $K$ such that both $U$ and $U/K$ are simply-connected, then can we conclude that $K$ is connected?
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### Moscow space-Examples

A space $X$ is called $Moscow$, if for each open subset $U$ of $X$, the closure of $U$ in $X$ is the union of a family of $Gâ€Ž_{Î´}$â€Žâ€Žâ€Ž â€Ž-subsets of $X$ . For example, Every first countable $T_1$ ...
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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 ...
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### Group theoretic lemma about the extension of homomorphisms of profinite groups

I have a question about a group-theoretic lemma proven in Galois Groups and Fundamental Groups by Tamas Szamuely. Suppose we have a profinite group $\Gamma$, a closed normal subgroup $N \subset \Gamma$...
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### Lie Groups: Identity Component

Main Problem Given a Lie group. The connected component of the identity is a Lie subgroup: It is a subgroup. It is open. How to check this using topological tools? Extra Problem The quotient ...
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### Prove that a subgroup in a Lie group is homogeneous

Let $\mathbb E:=(\mathbb R^4, \cdot)$ be a Carnot group whose Lie algebra is given by $\mathfrak g=V_1\oplus V_2 \oplus V_3$, where $V_1=span\{X_1,X_2\},$ $V_2=span\{X_3\},$ $V_3=span\{X_4\}$, the ...
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### Under what mild condition, inclusion of a conjugate subgroup in the initial subgroup gives equality?

Let $G$ be a group, $H \leq G$ a subgroup, and $x \in G$. If $xHx^{-1} \subset H$, then we may not have $xHx^{-1} = H$. Here is a simple counterexample. But this don't prevent the hope to finding ...
### The path-connected field is just $\mathbb{R}$ or $\mathbb{C}$?
A topological ring is a ring $R$ which is also a topological space such that both the addition and the multiplication are continuous as maps. $F$ is a topological field, if $F$ is a topological ring, ...