Generally, local invertibility does not imply invertibility. However, for differentiable functions from to 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 which are at least topological spaces (but may be more structured) and subsets of endomorphisms on for which local invertibility implies invertibility.
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