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Suppose $X,Y$ are Banach spaces and consider the space $C(X,Y)$ of bounded continuous functions $X \to Y$ with the supremum norm. Are there any results about the the topological properties of the set $K = \{f \in C(X,Y): f \text{ is bijective }\}$?

For instance, I would like to make statements such as

"If $c: (0,1) \to C(X,Y)$ is a curve satisfying some condition then there exists $t \in (0,1)$ such that $c(t) \in K$"

Although such a theorem would probably be too much to ask, this has motivated me to think about the properties of $K$ in general.

Thanks in advance.

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I've never seen bounded continuous functions on a Banach space come up before. Why is the structure of the space as a vector space relevant when we are considering nonlinear maps? – Alex Becker Feb 24 '12 at 20:54
How can a function be both bounded and bijective? Do you mean "bounded" in the sense of a bounded linear operator? – Jim Belk Feb 25 '12 at 6:02
OK sorry guys. I realize now that the question makes no sense at all. – user12014 Feb 25 '12 at 20:19

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