TITLE: When does local invertibility imply invertibility?
QUESTION [11 upvotes]: Generally, local invertibility does not imply invertibility.  However, for differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ then surjectivity and local invertibility do imply invertibility.
As well as being the most obvious, it's also the only (non-contrived) case that I can think of.  Are there any more?
Specifically, I'm looking for examples of spaces $X$ which are at least topological spaces (but may be more structured) and subsets of endomorphisms on $X$ for which local invertibility implies invertibility.

REPLY [5 votes]: Following Deane Yang, the answer is a definite yes: the map 
in question is a global diffeo,
provided that 
(a) it is `locally invertible': i.e. its   derivative is everywhere invertible,  and
(b) the domain and range are compact, simply connected, without boundary.
Proof: the map must be a covering map (``stack of records theorem'' -- see
for example Guillemin and Pollack). But a covering map between simply connected spaces
is an isomorphism -- here a diffeo. 
To make this `non-contrived' take domain and range to be the two-sphere.