Josh Mohanty gave me a math problem :D.
Suppose you have a topological space X. Let I(X) be the set of invertible mappings from X to itself. And let H(X) be the set of bicontinuous mappings from X to itself (the homeomorphism group). Bicontinuous means bijective and both the function and its inverse are continuous.
The questions.
- Is the inverse function F : I(X) \rightarrow I(X), where F(f) = f^{-1}, continuous on I(X) under the compact-open topology?
- What about when restricted to H(X)?
- What properties of X are required for the above two to be true?
I would guess the second claim is not true in general but probably true in metric spaces. I’m going to try to see if holds for \R under the usual topology first.
If you figure it out put it in a spoiler tag and give me a hint