Why do group actions necessarily imply homeomorphisms? In reading Hatcher (p135), I see that an element of a group acting on a space is a homeomorphism.  Working from the definition of group action, I don't see why this would necessarily be true.  Am I missing something obvious?
 A: EDIT: Note that what I call "$Aut(X)$," Hatcher calls "$Homeo(X)$."
It doesn't make sense to say "a group acting on a space is a homeomorphism". A homeomorphism is a single map, from one space to another (possibly the same space).
What is true is, that if $G$ acts on $X$, then for each $g\in G$ the map $$act_g: X\rightarrow X: x\mapsto gx$$ is a homeomorphism from $X$ to $X$.
Why is this? Well, by definition $act_g$ is a continuous map, so we just need to show that it has a continuous inverse. Consider $act_{g^{-1}}$. This is again a continuous map from $X$ to $X$, and we have $act_g\circ act_{g^{-1}}=id=act_{g^{-1}}\circ act_g$ (since $gg^{-1}x=(gg^{-1}x=ex=x=ex=(g^{-1}g)x=g^{-1}gx$).

There is a sense in which a group acting on a space is a single map: namely,


*

*An action of a group $G$ on a set $X$ is a group homomorphism $h: G\rightarrow Sym(X)$.

*An action of a group $G$ on a topological space $X$ is a group homomorphism $h: G\rightarrow Aut(X)$, where "$Aut(X)$" is the group of autohomeomorphisms of $X$ (that is, homeomorphisms from $X$ to itself).
So in this sense, we can speak of a group action as a single map, and ask whether or not it is an isomorphism of groups (usually it is not). However, if we think of the individual maps making up the group action - formally, the elements of the image of $h$ - then each of these is indeed a homeomorphism from $X$ to itself. I've shown above how to deduce this from the axioms of group actions; if you define a group action on a space as a group homomorphism to $Aut(X)$, though, this becomes even easier, since by definition any element of the image is an element of $Aut(X)$ and hence an autohomeomorphism.
