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.