Let $G$ be a compact Lie group acting effectively on a simply connected space $X$. Let Homeo$(X)$ be the group of all homeomorphisms of $X$ with itself given the compact open topology.
Is the topology on $G$ the same as the subspace topology it inherits from Homeo$(X)$? If not then is one topology finer than the other?
I'm not quite sure where to begin here. If I can somehow show that $G$ is closed in Homeo$(X)$ then I will be done. But I am unsure how to proceed (or even if it is true).
My ultimate goal is to see if $G$ is compact as a subspace of Homeo$(X)$. So even if I can show that the subspace topology on $G$ is contained in the topology of $G$ then it is enough (I'm not sure how to prove this either). However I was wondering if the two topologies are equal?