There is two definition of faithful action:

Definition 1: By a faithful action of a topological group $G$ on a topological manifold $M$, we mean a continuous injection $G \to \rm{Homeo}(M)$ (where $\rm{Homeo}(M)$ has the compact-open topology).

Definition 2: The action of $G$ on $M$ is called Faithful (or effective) if for every two distinct $g, h$ in $G$ there exists an $p\in M$ such that $g⋅p \neq h⋅p$.

How this two definitions of faithful action are related?


1 implies thas 2. if $f:G\rightarrow Homeo(M)$ is injective, for every $g\in G, f(g)\neq Id_M$ it implies that there exists $x,y\in M$ such that $f(g).x\neq f(g).y$

2 implies 1. Let $g\in G$ distinct of the identity, there exists $x,y\in M$ such that $f(g).x\neq f(g).y$ this implies that $f(g)\neq Id_M and $f$ is injective.

You can also say: 2. is equivalent to says that for every $g,h$ distinct, there exists $p\in M$ such that $f(g).p\neq f(h).p$ i.e $f(h^{-1}g).p\neq p$ i.e $f(h^{-1})g\neq Id_M$ and this means $f(g)\neq f(h)$.


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