Effective Topological Transformation Groups and the Group of Homeomorphisms

Question

I'm reading Steenrod's Topology of Fibre Bundles, and on pages 6 and 7, he defines a topological group $G$ and a topological transformation group of a topological space (which I understand to be a verbose way of saying a group action on a topological space). Now, he writes

For any fixed $g$, the map $y\mapsto g\cdot y$ is a homeomorphism of $Y$ onto itself; for it has a continuous inverse $y\mapsto g^{-1}\cdot y$. In this way the [group action] provides a homomorphism of $G$ into the group of homeomorphisms of $Y$.

We shall say that $G$ is effective if $g\cdot y=y$ for all $y$ implies $g=e$. Then $G$ is isomorphic to the group of homeomorphisms of $Y$.

It is this last statement (in bold) that I'm not sure about: it is clear than an effective group action implies the homomorphism from $G$ to the group of homeomorphisms of $Y$ is injective, but it is not at clear to me that this map is surjective.

Any ideas?

Answer 1

You misquoted the book. It actually says

Then $G$ is isomorphic to a group of homeomorphisms of $Y$.

At least that's how it reads in my edition (copyright 1951, ninth printing, 1974).