# Topology of bijective functions between Banach spaces

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.