1
vote
0answers
27 views

Connected component and group action

Let $G$ be a topological group acting on a set $X$. Let $x \in X$ and consider the orbit $G.x$ endowed with the topology coming from the quotient $G/ Stab(x)$. If $G^0$ is the connected component of ...
0
votes
0answers
36 views

Isometry groups acting transitively

Let $X$ be a metric space and $G$ be its group of isometries. 1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which ...
2
votes
0answers
39 views

Invariance of Decomposition of Invariant Functional

Let $Q$ a locally compact group acting on a locally compact space $X$ on the left. Let $\mathcal{A}$ a Banach space of bounded continuous functions $f:X\to\mathbb{C}$ and $m\in\mathcal{A}^{\ast}$ a ...
5
votes
1answer
108 views

A question about quotient under group action

Let $X$ be a Hausdorff space, and $G$ a group acting on $X$ by homeomorphisms. Let $H$ be a normal subgroup of $G$. Is it true that $X/G$ is homeomorphic to $(X/H)/(G/H)$ ? If so, can you please ...
6
votes
1answer
359 views

wiki's definition of “strongly continuous group action” wrong?

Wikipedia defines strongly continuous group action as follows: A group action of a topological group G on a topological space X is said to be strongly continuous if for all x in X, the map g ↦ ...
1
vote
0answers
39 views

Finiteness of fixed points of a Lie group action

Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$. Under which assumptions there will be a ...
5
votes
1answer
502 views

Quotient of a locally compact Hausdorff space by a proper action is Hausdorff

I am trying to prove the following: Let $G$ be a topological group acting properly on a Hausdorff locally compact space $X$, i.e. preimages of compacts sets by the map $$G\times X\to X\times ...
4
votes
1answer
133 views

Continuous Actions and Homomorphisms

I am learning about the compact-open topology and have a small proposition I am struggling to prove. Let $G$ be a topological group, $X$ a compact, Hausdorff space, and $H(X)$, the homeomorphisms of ...