# Action of a compact Lie group is a proper action

I want to show that the usual action of $$U(1)$$ on $$S^3$$ given by $$e^{i\theta}\cdot(z_1,z_2)=(e^{i\theta}z_1,e^{i\theta}z_2)$$ is a proper action.

It seems that the following more general statement should be true

Any action of a compact Lie group is a proper action

Recall that the action of a topological group $$G$$ on $$X$$ is a map $$G\times X\rightarrow X$$ given by $$(g,x)=g\cdot x$$. The action is said to be proper if the map $$G\times X \rightarrow X \times X$$ given by $$(g,x)\mapsto(g\cdot x,x)$$ is a proper, that is the preimage of every compact set is compact.

I have having trouble proving this: Let $$U\subset X\times X$$ be compact.The projections $$\pi_1,\pi_2:X\times X\rightarrow X$$ onto the first and second coordinates are continuous. Therefore $$\pi_1(U)$$ and $$\pi_2(U)$$ are compact in $$X$$.

I am unsure what the preimage of $$U$$ is however. Obviously it is $$\{(g,x)\in G \times X \text{ such that }(g\cdot x,x) \in U\}$$ But I do not know how to continue.

Let $$f:G\times X\rightarrow X\times Y$$ defined by $$f(g,x)=(gx,x)$$ and $$U$$ be a compact subset of $$X\times X$$, $$\pi_2(U)=V$$ is compact since $$\pi_2$$ continuous. $$f^{-1}(U)\subset G\times V$$ and is closed thus it is compact since it is a closed subset of a compact set.
Knowing that the action is continuous then the orbit map $$\mathcal{O}_x:G\to X$$ is continuous. Therefore assuming $$U\subseteq X$$ is compact and $$X$$ is Hausdorff (this is generally assumed for most applications, but please specify if this is not allowed) then we find that $$U$$ is closed in $$X$$. Therefore $$\mathcal{O}_x^{-1}(U) \subseteq G$$ is closed, and thus compact since $$G$$ is compact. Therefore $$\mathcal{O}_x^{-1}(U)\times U$$ will be compact in $$G\times X$$.
• This is good enough for what I am doing since since $S^3$ is Hausdorff. Out of curiosity and for the benefit other people who make see this question in the future: Is the statement true if $X$ is not Hausdorff?
• @Milkman From what I've seen in the literature, proper actions are generally only defined for Hausdorff or locally Hausdorff sets $X$. I'm not sure it's technically defined for arbitrary topological spaces. The main reason why I see this being an issue is that getting rid of the Hausdorff condition may generally yield pathological examples where we don't get nice quotients. The only theorem which I feel is more general is the homogeneous space construction theorem where the group action needs to be transitive, but it can act on an arbitrary set. Commented Dec 1, 2019 at 0:19