Continuous Actions of Groups

Subjects: Group Theory, Topology
Links: Topological Groups, Group Actions, Quotient Topology, Proper Maps

Def: Suppose $X$ is a topological space and $G$ is a group action on $X$. The action is called an action by homeomorphism if for each $g\in G$, the map $x\mapsto g\cdot x$ is a homeomorphism of $X$.

Def: Suppose an action of a topological group $G$ on a topological space $X$, $\alpha :G\times X\to X$ is said be a continuous action if $\alpha$ is continuous with the product topology. We call $X$ a $G$-space.

Prop: Suppose $G$ is a topological group acting a a topological space $X$.

• If the action is continuous, then it is an action by homeomorphism.
• If $G$ has a discrete topology, then the action is continuous iff it is an action by homeomorphism.

We say two points $x,y\in X$ are equivalent if they are in the same orbit, i.e., there is an element $g\in G$, such that $y=\alpha (g,x)$. Let $X/G$ be the quotient space of this equivalence relation, called the orbit space of the action $\alpha$.

Prop: The map $\pi :X\to X/G$ is an open map. Meaning that the equivalence relation is open.

Hausdorff Criterion for Orbit Spaces: Suppose $E$ is a topological space and $\mathrm{\Gamma }$ is a group acting on $E$ by homeomorphisms. The following statements are equivalent:

• Then $E/\mathrm{\Gamma }$ is Hausdorff.
• If $e,e^{\prime }\in E$ lie in different orbits, there exists neighbourhoods $U$ of $e$ and $V$ of $e^{\prime }$ such that $U\cap (g\cdot V)=\varnothing$ for all $g\in \mathrm{\Gamma }$.