In a paper an author proved the following proposition
Please help me in trace proof of following proposition
Proposition: let $f$ be a homeomorphism of a connected topological manifold $M$ with fixed point set $F$. then either $(1)$ $f$ is invariant on each component of $M-F$ or $(2)$ there are exactly two component and $f$ interchanges them.
and after that he said:
In the case of $(2)$ the above argument shows that F cannot contain an open set and hence $dim F\leq (dim M) -1$ and since $F$ separates $M$ we have $dim F = (dim M) -1$. G. Bredon has shown that if $M$ is also orientable then any involution with an odd codimensional fixed point set must reverse the orientation; hence we obtain
Let $f$ be an orientation-preserving homemorphism of an orientable manifold $M$; then $f$ is invariant on each component of $M-F$.
Can you say me, what does mean the dim $F$ here? Is always $F$ is sub manifold with above condition? and how can we deduce that $dim F = n-1$?