Let $G$ be a compact Lie group acting effectively on a compact, Hausdorff topological space $X$. I am looking for results of the type 


*If $X$ is a ... and the action is ... then $\dim(X/G)\leq \dim(X)-\dim(G)$*.


Here $\dim$ denotes the [covering dimension](http://en.wikipedia.org/wiki/Lebesgue_covering_dimension).


For example, $X$ is a smooth manifold and $G$ acts smoothly and almost-freely (all isotropy groups finite).


Can anyone point me to a more general theorem of this type?



Dimensions of orbit spaces
I'm loathe to answer my own question, but...


Theorem IV.3.8 of Bredon's book [*Introduction to compact transformation groups*](http://books.google.co.uk/books?id=qXEl_AIeSBUC&printsec=frontcover&dq=introduction+to+compact+transformation+groups&source=bl&ots=w_Rpcfjv_F&sig=C8MiAvvNNsAwgf7Ki3VUixSe3Zk&hl=en&ei=nrtJTeviGsWYhQew9NjoDg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBYQ6AEwAA#v=onepage&q&f=false) (which sadly is on the page google books decided not to include!) seems satisfying.


Let $G$ be a compact Lie group acting locally smoothly on the manifold $M$, such that $M/G$ is connected. If $P$ is a principal orbit (an orbit of maximum dimension) then


$$\dim(M/G)=\dim(M)-\dim(P).$$


